A simplicial complex spliting associativity

TL;DR

This paper introduces a new family of simplicial objects called $ ext{Dy}^m$ operads, generalizing associative and dendriform algebras, with dimensions given by Fuss-Catalan numbers, and constructs $ ext{Dy}^m$ algebras from combinatorial posets.

Contribution

It defines a novel simplicial object in non-symmetric operads that unifies associative and dendriform structures, extending to a family of $ ext{Dy}^m$ operads with Fuss-Catalan dimensions.

Findings

$ ext{Dy}^m$ operads generalize associative and dendriform operads.

Dimensions of $ ext{Dy}^m$ are given by Fuss-Catalan numbers.

Construction of $ ext{Dy}^m$ algebras from combinatorial posets.

Abstract

We introduce a simplicial object $(\{ \Dy^m\}_{m\geq 0}, {\mathbb F}_i, {\mathbb S}_j)$ in the category of non-symmetric algebraic operads, satisfying that $\Dy^0$ is the operad of associative algebras and $\Dy^1$ is J.-L. Loday\rq s operad of dendriform algebras. The dimensions of the operad $\Dy^m$ are given by the Fuss-Catalan numbers. Given a family of partially ordered sets ${\bold P}=\{P_n\}_{n\geq 1}$ we show that, under certain conditions, the vector space spanned by the set of $m$-simpleces of ${\bold P}$ is a $\Dy^m$ algebra. This construction, applied to certain combinatorial Hopf algebras, whose associative product comes from a dendriform structure, provides examples of $\Dy^m$ algebras.

Equations

$$x{\ast }_i(y{\ast }_{j}z)=(x{\ast }_{i}y){\ast }_{j}z,\mathrm{for}0\le i<j\le m,$$

$$\sum _{j=0}^{i}x{\ast }_i(y{\ast }_{j}z)=\sum _{k=i}^m(x{\ast }_{k}y){\ast }_{i}z,\mathrm{for}0\le i\le m,$$

$$x{\circ }_i(y{\circ }_{j}z)-(x{\circ }_{i}y){\circ }_{j}z=x{\circ }_i(y{\circ }_{j-1}z)-(x{\circ }_{i}y){\circ }_{j-1}z,$$

$$x{\circ }_i(y{\circ }_{i}z)=(x{\circ }_{m}y){\circ }_{i}z-(x{\circ }_{m}y){\circ }_{i-1}z+x{\circ }_{i-1}(y{\circ }_{i-1}z),$$

$$\text{mbox}{Dyck}^m(V):=\underset{n\ge 1}{⨁}\text{mbox}{Dyck}_n^m\otimes V^{\otimes n},$$

$$t\otimes (v_1\otimes \cdots \otimes v_n){\ast }_{i}w\otimes (u_1\otimes \cdots \otimes u_p):=(t{\ast }_{i}w)\otimes (v_1\otimes \cdots \otimes v_n\otimes u_1\otimes \cdots \otimes u_m),$$

$$(x{\ast }_{i}y){\ast }_{i}z=\sum _{j=0}^{i}x{\ast }_i(y{\ast }_{j}z)-\sum _{j=i+1}^m(x{\ast }_{j}y){\ast }_{i}z,\mathrm{for}0\le i\le m,$$

$$t{\ast }_{i}w=\sum _{k=0}^i{t}^l{\ast }_i(t^r{\ast }_{k}w)-\sum _{k=i+1}^m(t^l{\ast }_k{t}^r){\ast }_{i}w.$$

$$t{\ast }_{i}w:=\sum _{k=0}^i{t}^l{\vee }_i(t^r{\ast }_{k}w)-\sum _{k=i+1}^m(t^l{\ast }_k{t}^r){\vee }_{i}w.$$

$$\tilde{f}(t)=\tilde{f}(t^l){\ast }_i^D\tilde{f}(t^r).$$

$${\mathcal{B}}^m(V):=\bigcup _{n\ge 1}{\mathcal{B}}_n^m\times X^n,$$

$$d_{m,n}:=\frac{1}{mn+1}\left(\genfrac{}{}{0ex}{}{(m+1)n}{n}\right),$$

$$x\cdot d_m(x{)}^{m+1}=d_m(x)-1.$$

$${\Omega }_{i_1,\dots ,i_p}^L(t_1,\dots ,t_p):=(((|{\vee }_{i_p}{t}_p){\vee }_{i_{p-1}}{t}_{p-1})\dots ){\vee }_{i_1}{t}_1,$$

$${\Omega }_{i_1,\dots ,i_p}^R(t_1,\dots ,t_p):=t_1{\vee }_{i_1}(t_2{\vee }_{i_2}(\dots (t_{p-1}{\vee }_{i_{p-1}}(t_p{\vee }_{i_p}|)))).$$

$$t={\Omega }_{i_1,\dots ,i_p}^L(t_1,\dots ,t_p)={\Omega }_{j_1,\dots ,j_q}^R(w_1,\dots ,w_q),$$

$$t^l={\Omega }_{i_2,\dots ,i_p}^L(t_2,\dots ,t_p)=w_1\mathrm{and}{t}^r=t_1={\Omega }_{j_2,\dots ,j_q}^R(w_2,\dots ,w_q).$$

$$f_m(x):=\sum _{n\ge 1}{b}_{m,n}{x}^n,$$

$$t={\Omega }_{i_1,\dots ,i_p}^L(t_1,\dots ,t_p).$$

$$\displaystyle\sum_{1\leq p\leq{n-1}}\bigl{(}\sum_{0\leq i_{1}<\dots<i_{p}\leq m}$$

$$\sum _{1\le p\le n-1}\left(\genfrac{}{}{0ex}{}{m+1}{p}\right)(\sum _{n_1+\cdots +n_p=n-1}{b}_{m,n_1}\cdot \dots \cdot b_{m,n_p}).$$

$$x(1+f_m(x){)}^{m+1}=$$

$$\displaystyle x\bigl{(}\sum_{r=0}^{m+1}\binom{m+1}{r}$$

$$\displaystyle\sum_{r=0}^{m+1}\binom{m+1}{r}\bigl{(}\sum_{n_{1},\dots,n_{r}}b_{m,n_{1}}\cdot\ldots\cdot b_{m,n_{p}}x^{n_{1}+\dots+n_{r}+1}\bigr{)}.$$

$$\sum_{r=0}^{m+1}\binom{m+1}{r}\bigl{(}\sum_{n_{1}+\dots+n_{r}+1=n}b_{m,n_{1}}\cdot\ldots\cdot b_{m,n_{p}}\bigr{)},$$

$$((x{\ast }_{0}y){\ast }_{0}z+(x{\ast }_{0}y){\ast }_{1}z)=-a(x{\ast }_{1}y){\ast }_{0}z-b(x{\ast }_{1}y){\ast }_{1}z,$$

$$(x{\bullet }_{i}y){\bullet }_{i+j}z=x{\bullet }_i(y{\bullet }_{j}z),$$

$${\underline{\ast }}_j:=\{\begin{array}{ll}{\ast }_j,& \mathrm{for}0\le j\le i\text{mbox}-1,\\ 0,& \mathrm{for}j=i,\\ {\ast }_{j-1},& \mathrm{for}i+1\le j\le m+1.\end{array}$$

$${\overline{\ast }}_j:=\{\begin{array}{ll}{\ast }_j,& \mathrm{for}0\le j<i,\\ {\ast }_i+{\ast }_{i+1},& \mathrm{for}j=i,\\ {\ast }_{j-1},& \mathrm{for}i+1<j\le m,\end{array}$$

$${\Omega }_{j_{l_0},\dots ,j_p}^R({\Omega }_{s_1,\dots ,s_q,k,r_1,\dots ,r_h}^L(w_1,\dots ,w_q,w_{q+1},t_1^l,\dots ,t_h^l),t_{j_{l_0+1}}^r,\dots ,t_p^r),$$

$$\Theta (t{)}^l:=\Theta (\Omega$$

$${\Omega }_{s_1,\dots ,s_q,k,r_1,\dots ,r_h}^L(\Theta (w_1),\dots ,\Theta (w_{q+1}),\Theta (t_1^l),\dots ,\Theta (t_h^l)).$$

Full text

Given a family of partially ordered sets ${\mathbb{P}}=\{P_{n}\}_{n\geq 1}$ we show that, under certain conditions, the vector space spanned by the set of $m$-simpleces of ${\mathbb{P}}$ is a $\mbox{\it Dyck}^{m}$ algebra. This construction, applied to certain combinatorial Hopf algebras, whose associative product comes from a dendriform structure, provides examples of $\mbox{\it Dyck}^{m}$ algebras.

Our joint work was partially supported by the projects Fondecyt Postdoctorado 3140298, Fondecyt Regular 1171209 and MathAmSud 17-Math-05 LIETS.

Introduction

In [28], J.-L. Loday introduced the notion of dendriform algebra as a type of associative algebra, whose product splits as the sum of two binary operations. Many associative algebras, as the algebras defined by shuffles (see [16], [29]) and the Rota-Baxter algebras (see [1]), are examples of dendriform algebras.

Recent publications (see for instance [23], [10], [2], [3] and [21]) deal with different ways of spliting associativity, involving the dendriform operad. We introduce a family of non-symmetric operads $\{\mbox{\it Dyck}^{m}\}_{m\geq 0}$, equipped with operad morphisms ${\mathbb{F}}_{i}:\mbox{\it Dyck}^{m}\longrightarrow\mbox{\it Dyck}^{m-1}$ and ${\mathbb{S}}_{i}:\mbox{\it Dyck}^{m}\longrightarrow\mbox{\it Dyck}^{m+1}$, for $0\leq i\leq m$, satisfying that:

1. (1)

$\mbox{\it Dyck}^{0}$ is the operad As of associative algebras and $\mbox{\it Dyck}^{1}$ is the operad of dendriform algebras, 2. (2)

$(\{\mbox{\it Dyck}^{m}\}_{m\geq 0},{\mathbb{F}}_{i},{\mathbb{S}}_{j})$ is a simplicial object in the category of non-symmetric operads, 3. (3)

the dimension of the subspace of homogeneous elements of degree $n$ of the operad $\mbox{\it Dyck}^{m}$ is the Fuss-Catalan number $d_{m,n}$.

As $\mbox{\it Dyck}^{m}$ is a non-symmetric operad, it is completely described by its free object over one element (see [34] and [32]).

For $m=1$, the Fuss-Catalan number $d_{1,n}$ coincides with the Catalan’s number $c_{n}=\frac{1}{n+1}\binom{2n}{n}$, which is the cardinal of the set ${\mathcal{Y}}_{n}$ of planar binary rooted trees with $n{\mbox{+}1}$ leaves. In [29], the free dendriform algebra over one element was described on the vector space ${\mathbb{K}}[{\mathcal{Y}}]:=\bigoplus_{n\geq 1}{\mathbb{K}}[{\mathcal{Y}}_{n}]$, spanned by the set of planar binary rooted trees. In [30], J.-L. Loday and the third author proved that the dendriform structure of ${\mathbb{K}}[{\mathcal{Y}}]$, is completely determined by the Tamari order $\leq_{Ta}$ (see [45]), and two morphisms of partially ordered sets $/,\backslash:{\mathcal{Y}}_{n}\times{\mathcal{Y}}_{r}\longrightarrow{\mathcal{Y}}_{n+r}$, for $n,r\geq 1$.

The last result also holds for other dendriform algebras, whose underlying vector spaces admit a graded basis $\bigcup_{n\geq 1}P_{n}$, where $P_{n}$ is a partially ordered set, for $n\geq 1$. Let us mention the dendriform algebras spanned by

1. (1)

the sets $\Sigma_{n}$ of permutations of $n$ elements equipped with the weak Bruhat order (see [30]), 2. (2)

the sets of surjective maps ${\mbox{Surj}_{n}}$, equipped with the facial order introduced in [26]. Dendriform structures on the vector space spanned by the set $\bigcup_{n\geq 1}{\mbox{Surj}_{n}}$, were defined in [12] and in [31]. 3. (3)

the sets of planar rooted trees ${\mathcal{T}}_{n}$ , equipped with the partial order defined in [38], which extends the Tamari order.

The functor ${\mbox{Simp}(P)}$ associates to any partially ordered set $(P,\leq)$ a simplicial set ${\mbox{Simp}(P)}$, whose $m$-simpleces are the $m$-tuples $(p_{1}\leq\dots\leq p_{m})$ in $P^{m}$. Given a family ${\mathbb{P}}=\{P_{n}\}_{n\geq 1}$ of partially ordered sets and $m\geq 1$, we denote by ${\mbox{Simp}({\mathbb{P}})^{m}}$ the graded set of $m$-simpleces, whose elements of degree $n$ are the $m$-simpleces of $P_{n}$, for $n\geq 1$. In Section 3, we introduce the notion of dendriform poset. The vector space spanned by a dendriform poset is a dendriform algebra, while the vector space spanned by the set ${\mbox{Simp}({\mathbb{P}})^{m}}$, of $m$-simpleces of $\{\mathbb{P}\}$, has a natural structure of $\mbox{\it Dyck}^{m}$ algebra. This result provides examples of $\mbox{\it Dyck}^{m}$ algebras. For instance, we prove that the families of partially ordered sets $\{{\mathcal{Y}}_{n}\},\{\Sigma_{n}\},\{{\mbox{Surj}_{n}}\}$ and $\{{\mathcal{T}}_{n}\}$ described in the paragraph above, are dendriform posets.

On the other hand, motivated by the combinatorics of the Garsia-Haiman spaces, F. Bergeron introduced in [6] a generalization of the Tamari lattice, called the $m$-Tamari lattice, defined on the sets of $m$-Dyck paths. In the last section, we describe a $\mbox{\it Dyck}^{m}$ algebra structure on the vector space spanned by the set $\bigcup_{n\geq 1}\mbox{\it Dy}_{n}^{m}$, of all $m$-Dyck paths. We prove that the products $*_{0},\dots,*_{m}$, which define the $\mbox{\it Dyck}^{m}$ algebra, are described by intervals of the $m$-Tamari order.

In [37], J.-C. Novelli and J.-Y. Thibon introduced the combinatorial Hopf algebra ${}^{m}{\mbox{\bf{FQSym}}}$ of $m$-permutations. Their construction provide $m$-analogues of a large family of combinatorial Hopf algebras, in particular they defined a combinatorial Hopf algebra ${}^{m}{\mbox{\bf{PBT}}}$, whose underlying vector space is spanned by the set of planar rooted $m$-ary trees, and proved that its associative product is described by the $m$-Tamari lattice. J.-C. Novelli introduced in [35] a family of operads, called $m$-dendriform operads, whose dimensions are the Fuss-Catalan numbers. Though the operad of $m$-dendriform algebras and $\mbox{\it Dyck}^{m}$ have the same dimensions, both operads are not isomorphic for $m\geq 2$. Clearly, using any bijective map between the set of $m$-Dyck paths and the set of $m$-ary planar rooted trees, we may define a $\mbox{\it Dyck}^{m}$ algebra structure on the vector space spanned by the second set. But the description of this $\mbox{\it Dyck}^{m}$ algebra is much more complicated than the one defined on $m$-Dyck paths.

Our motivation to work on this type of algebraic operads is two fold.

1. (1)

Let ${\mathcal{P}}$ be an operad such that there exists, for $n\geq 1$, a partial order defined on a basis $P_{n}$ of the space ${\mathcal{P}}_{n}$, satisfying that these orders are compatible with the operad structure. An interesting problem is to study the existence of an operad ${\mbox{Ord}^{m}}({\mathcal{P}})$, spanned by the operations ${\mbox{Simp}({\mathbb{P}})^{m}}$, for $m\geq 1$, which must be a quotient of the Hadamard product ${\mathcal{P}}^{\otimes_{H}m}$. The dendriform operad Dend is an example of this type of operad, the set ${\mathcal{Y}}_{n}$ of planar rooted binary trees is a basis of ${\mbox{Dend}}_{n}$, and the Tamari order is compatible with the operad structure. Nevertheless, ${\mbox{dim}_{{\mathbb{K}}}(\mbox{\it Dyck}_{3}^{2})}=12$ and ${\mbox{dim}_{{\mathbb{K}}}({\mbox{Ord}^{2}}({\mbox{Dend}})_{3})}$ is $13$, so even if the operad ${\mbox{Ord}^{m}}({\mbox{Dend}})$ exists, it is not $\mbox{\it Dyck}^{m}$. In recent publications, as [18], [14], [39] and [40], algebraic structures related to families of partially ordered sets were described, some of them give new examples of dendriform algebras. 2. (2)

in [13], F. Chapoton introduced a differential non-symmetric operad ${\mathcal{K}}$, whose algebras are dendriform algebras equipped with an additional associative product satisfying certain relations. The operad ${\mathcal{K}}$ is Hopf, which implies that any free ${\mathcal{K}}$ algebra has a natural structure of conilpotent bialgebra. As proved in [11], the subspace of primitive elements of any ${\mathcal{K}}$ bialgebra is a ${\mathcal{S}}_{2}$ algebra, where ${\mathcal{S}}_{2}$ is the second filtration stage of the surjection operad ${\mathcal{S}}$ (see [5]), so the operad ${\mathcal{S}}_{2}$ is an $E_{2}$ operad. Moreover, there exists an equivalence between the category of conilpotent ${\mathcal{K}}$ bialgebras and the category of ${\mathcal{S}}_{2}$-algebras, which implies that the operad ${\mathcal{S}}_{2}$ is completely described in terms of the operad ${\mathcal{K}}$ (see [11]). In [4], M. Batanin and C. Berger introduced the filtered colored operad ${\mathcal{L}}$, called the lattice path operad, whose $n^{th}$ filtration stage ${\mathcal{L}}_{n}$ is an $E_{n}$ operad, for $n\geq 1$. Our aim is to study non-symmetric Hopf operads ${\mathcal{P}}$ defined on the space spanned by all lattice paths, applying the ideas developed by X. Viennot and the second author in [43]. As the operad ${\mathcal{L}}$ is symmetric, we are interested in the properties of the operads defined on the subspaces of primitive elements of ${\mathcal{P}}$ bialgebras and in their relationship with the operad ${\mathcal{L}}$.

Contents

The first section contains the definition of $\mbox{\it Dyck}^{m}$ algebras and the description of the free $\mbox{\it Dyck}^{m}$ algebra over one element. We prove that the dimensions of $\mbox{\it Dyck}^{m}$ are given by the Fuss-Catalan numbers, and compare the operad $\mbox{\it Dyck}^{2}$ to other operads having the same dimensions, in particular with the operad of $m$-dendriform algebras introduced in [35].

The simplicial object $(\{\mbox{\it Dyck}^{m}\}_{m\geq 0},{\mathbb{F}}_{i},{\mathbb{S}}_{j}\})$ is described in Section 2. We show that the degeneracy functors ${\mathbb{S}}_{j}:{\mbox{}\mbox{\it Dyck}^{m+1}{\mbox{-alg}}}\longrightarrow{\mbox{}\mbox{\it Dyck}^{m}{\mbox{-alg}}}$ preserve free objects.

In Section 3 we introduce the definition of dendriform poset, in such a way that the vector space spanned by any dendriform poset is a dendriform algebra. We prove that, for any dendriform poset ${\mathbb{P}}$, the vector space ${\mbox{Ord}^{m}}({\mbox{\bf P}})$ spanned by the $m$-simpleces of the partially ordered sets $P_{n}$, is a $\mbox{\it Dyck}^{m}$ algebra. Using previous results, we describe examples of $\mbox{\it Dyck}^{m}$ algebras coming from dendriform posets.

We recall basic definitions and constructions of $m$-Dyck paths, and define binary products $*_{0},\dots,*_{m}$ on the space ${\mathbb{K}}[\mbox{\it Dy}^{m}]$, spanned by the set of $m$-Dyck paths, in Section 4. We prove that the operad $\mbox{\it Dyck}^{m}$ is entirely described by the data $({\mathbb{K}}[\mbox{\it Dy}^{m}],*_{0},\dots,*_{m})$, so the combinatorial properties of $m$-Dyck paths define completely the operad. Finally, we show the $m+1$ binary operations $*_{0},\dots,*_{m}$ are given by intervals of F. Bergeron’s $m$-Tamari lattice.

Acknowledgements

D. López N. and M. Ronco want to thank specially Prof. Antonio Laface for his interest and support. The second author would like to thank Luc Lapointe for many fruitful discussions.

Preliminaries

All the vector spaces considered in the present work are over ${\mathbb{K}}$, where ${\mathbb{K}}$ is a field. For any set $X$, we denote by ${\mathbb{K}}[X]$ the vector space spanned by $X$. For any positive integer $n\geq 1$, we denote the set $\{1,\dots,n\}$ by $[n]$.

1. $\mbox{\it Dyck}^{m}$ algebras

1.1. Definition and basic properties

Dendriform algebras were introduced by J.-L. Loday in [28]. However, the first example of this type of algebra appeared many years before in [16], S. Eilenberg and S. MacLane introduced the shuffle product on simplicial complexes, using the Alexander-Whitney map, and defined a half product, denoted $\uparrow$, which splits it. In [1], M. Aguiar proved that Rota-Baxter algebras have a natural structure of dendriform algebras.

Definition 1.1.1**.**

(see [28]) A dendriform algebra over ${\mathbb{K}}$ is a vector space $A$ equipped with binary operations $\succ$ and $\prec$ satisfying the following conditions

1. (1)

$x\succ(y\succ z)=(x\succ y+x\prec y)\succ z$, 2. (2)

$x\succ(y\prec z)=(x\succ y)\prec z$, 3. (3)

$x\prec(y\succ z+y\prec z)=(x\prec y)\prec z$,

for $x,y,z\in A$.

Remark 1.1.2**.**

Suppose that we split $\succ$ as the sum of two binary operations $*_{0}$ and $*_{1}$. In order to describe new relations, in such a way that conditions $(1)$ and $(2)$ of Definiton 1.1.1 are satisfied, we set

1. (1)

$x*_{0}(y*_{0}z)=(x*y)*_{0}z$, 2. (2)

$x*_{0}(y*_{1}z)=(x*_{0}y)*_{1}z$, 3. (3)

$x*_{1}(y\succ z)=(x*_{1}y+x\prec z)*_{1}y$, 4. (4)

$x*_{i}(y\prec z)=(x*_{i}y)\prec z$, for $i=0,1$.

Note that the first three relations split relation $(1)$ of Definition 1.1.1, while the last relation splits the second one.

It is not difficult to verify that, if we perform an analogous procedure with $\prec$, that is we write $\prec=\circ_{1}+\circ_{2}$ and splits the relations of dendriform, we get the same type of algebra, with $3$ binary products.

Remark 1.1.2 motivates the following definition.

Definition 1.1.3**.**

For $m\geq 1$, a $\mbox{\it Dyck}^{m}$* algebra* over ${\mathbb{K}}$ is a vector space $D$ equipped with $m+1$ binary operations $*_{i}:D\otimes D\longrightarrow D$, for $0\leq i\leq m$, satisfying the following relations

[TABLE]

[TABLE]

for any elements $x,y$ and $z$ in $D$.

Let As denotes the operad of associative algebras. It is immediate to see that $\mbox{\it Dyck}^{0}={\mbox{As}}$. A $\mbox{\it Dyck}^{1}$ algebra is a dendriform algebra, for $\succ:=*_{0}$ and $\prec:=*_{1}$.

Any $\mbox{\it Dyck}^{m}$ algebra $(D,*_{0},\dots,*_{m})$ is an associative algebra, with the product $*:=*_{0}+\dots+*_{m}$. It is immediate to verify that the vector space $D$, with the products $\succ^{k}:=*_{0}+\dots+*_{k}$ and $\prec^{k}:=*_{k+1}+\dots+*_{m}$ is a dendriform algebra, for any $-1\leq k\leq m$.

Dendriform algebras cannot have units, see for instance [42]. A straightforward calculation proves that this result is also true for $\mbox{\it Dyck}^{m}$ algebras in general.

Remark 1.1.4**.**

Let $(D,*_{0},\dots,*_{m})$ be a $\mbox{\it Dyck}^{m}$ algebra. Define operations $\circ_{i}:=*_{0}+\dots+*_{i}$, for $0\leq i\leq m$.

It is easy to verify that conditions 1.1.1 and 1.1.2 are equivalent to

1. (1)

for $0\leq i<j\leq m$,

[TABLE] 2. (2)

$x\circ_{0}(y\circ_{0}z)=(x\circ_{m}y)\circ_{0}z$, 3. (3)

for $1\leq i\leq m$,

[TABLE]

for $x,y$ and $z$ in $D$.

1.2. The free $\mbox{\it Dyck}^{m}$ algebra

Remark 1.2.1**.**

Let $\mbox{\it Dyck}^{m}=\bigoplus_{n\geq 1}{\mbox{\it Dyck}_{n}^{m}}$ denote the free $\mbox{\it Dyck}^{m}$ algebra over one generator. As the operad $\mbox{\it Dyck}^{m}$ is non-symmetric, the free $\mbox{\it Dyck}^{m}$ algebra over a vector space $V$ (see [34] and [32]) is the vector space

[TABLE]

with the products

[TABLE]

for $t\in\mbox{\it Dyck}_{n}^{m}$, $w\in\mbox{\it Dyck}_{p}^{m}$, $v_{1}\otimes\dots\otimes v_{n}\in V^{\otimes n}$ and $u_{1}\otimes\dots\otimes u_{p}\in V^{\otimes p}$.

Remark 1.2.2**.**

In Definition 1.1.3 the equation 1.1.2 may be written as

[TABLE]

for $x,y$ and $z$ in $D$.

Notation 1.2.3**.**

For $n\geq 1$, let ${\mathcal{Y}}_{n-1}^{m}$ be the set of all planar binary rooted trees with $n$ leaves and the internal vertices colored by the elements of $\{0,\dots,m\}$. The unique element of ${\mathcal{Y}}_{0}^{m}$ is the tree $|$, with one leaf and no internal vertex.

Given two colored trees, $t$ and $w$, and an integer $0\leq i\leq m$, we denote by $t\vee_{i}w$ the colored tree obtained by connecting the roots of $t$ and $w$ to a new root colored by $i$, where $t$ is on the left side and $w$ on the right side of $t\vee_{i}w$.

For any internal vertex $v$ of a colored planar binary rooted tree $t\in{\mathcal{Y}}_{n-1}^{m}$, we denote by $t_{v}$ the colored subtree of $t$ whose root is $v$.

Note that any tree $t\in{\mathcal{Y}}_{n-1}^{m}$ may be written as $t=t^{l}\vee_{i}t^{r}$, for a unique integer $0\leq i\leq m$ and unique trees $t^{l}\in{\mathcal{Y}}_{n_{1}}^{m}$, $t^{r}\in{\mathcal{Y}}_{n_{2}}^{m}$ such that $n_{1}+n_{2}=n{\mbox{-}2}$.

Definition 1.2.4**.**

For $n\geq 2$, let ${\mathcal{B}}_{n}^{m}$ be the subset of all the elements $t$ in ${\mathcal{Y}}_{n-1}^{m}$ such that for any any subtree $t_{v}=t_{v}^{l}\vee_{i}t_{v}^{r}$ of $t$, the color of the root of $t_{v}^{l}$ is $j$, for some $j>i$.

For $n=1$, ${\mathcal{B}}_{1}^{m}={\mathcal{Y}}_{0}^{m}$ is the set whose unique element is the tree $|$.

Let ${\mathcal{B}}^{m}$ be the graded set $\bigcup_{n\geq 1}{\mathcal{B}}_{n}^{m}$. For any $t=t^{l}\vee_{i}t^{r}\in{\mathcal{B}}^{m}$, the trees $t^{l}$ and $t^{r}$ belong to ${\mathcal{B}}^{m}$.

Definition 1.2.5**.**

Let ${\mathfrak{D}}^{m}$ be the graded vector space whose basis is the set $\bigcup_{n\geq 1}{\mathcal{B}}_{n}^{m}$.

For any pair of trees $t\in{\mathcal{B}}_{n}^{m}$ and $w\in{\mathcal{B}}_{r}^{m}$, and any integer $0\leq i\leq m$, the product $t*_{i}w$ is defined recursively as follows,

1. (1)

for ${\mbox{}n=r=1}$, we set $|*_{i}|:=|\vee_{i}|$. 2. (2)

for $t=t^{l}\vee_{j}t^{r}$, or $n=1$, with $i<j\leq m$, define $t*_{i}w:=t\vee_{i}w\in{\mathcal{B}}_{n+r}^{m}$, 3. (3)

suppose that the products $t^{\prime}*_{i}w^{\prime}$ are defined for any pair of trees $t^{\prime}\in{\mathcal{B}}_{n_{1}}^{m}$ and $w^{\prime}\in{\mathcal{B}}_{r_{1}}^{m}$, for $n_{1}+r_{1}<n+r$. Moreover, we may also assume that, if $t^{\prime}*_{i}w^{\prime}=\sum_{u}\alpha_{t^{\prime}w^{\prime}}^{u}u$, then, for any $u\in{\mathcal{B}}_{n_{1}+r_{1}}^{m}$ such that $\alpha_{t^{\prime}w^{\prime}}^{u}\neq 0$, the color of the root of $u$ is largest or equal than the minimal element of the set $\{i,s\}$, where $s$ is the color of the root of $t^{\prime}$.

Let $t=t^{l}\vee_{j}t^{r}$, with $0\leq j\leq i$.

1. (a)

for $j<i$, define $t*_{i}w:=t^{l}\vee_{j}(t^{r}*_{i}w)$. 2. (b)

for $j=i$, applying Remark 1.2.2, we get

[TABLE]

Moreover,

$(i)$ as $t\in{\mathcal{B}}_{n}^{m}$, the color $s$ of the root of $t^{l}$ satisfies that $i<s$. So, the element $\sum_{k=0}^{i}t^{l}\vee_{i}(t^{r}*_{k}w)$ belongs to ${\mathfrak{D}}^{m}$,

$(ii)$ the product $t^{l}*_{k}t^{r}$ is a sum of trees, satisfying that the colors of their roots are largest or equal to ${\mbox{min}\{k,s\}}$, where $s$ is the color of the root of $t^{l}$. As both integers are largest than $i$, the roots of all trees appearing in $t^{l}*_{k}t^{r}$ are largest than $i$.

So, the element $t*_{i}w$ is defined as

[TABLE]

It is easily seen that the products $*_{i}$ are defined on ${\mathfrak{D}}^{m}$ in such a way that $({\mathfrak{D}}^{m},*_{0},\dots,*_{m})$ is a $\mbox{\it Dyck}^{m}$ algebra.

Example 1.2.6**.**

Let

we get that

The following result is a consequence of Remark 1.2.2 and Definition 1.2.5.

Proposition 1.2.7**.**

The graded vector space ${\mathfrak{D}}^{m}$, equipped with the products $*_{0},\dots,*_{m}$, is the free $\mbox{\it Dyck}^{m}$ algebra over one element $\#$.

• Proof. The linear homomorphism ${\mathbb{K}}\#\longrightarrow{\mathfrak{D}}^{m}$, maps $\#\mapsto|$.

Given a $\mbox{\it Dyck}^{m}$ algebra $(D,*_{0}^{D},\dots,*_{m}^{D})$, a linear map $f$ from ${\mathbb{K}}\#$ to $D$ is completely determined by the element $f(\#)\in D$. There exists a unique linear homomorphism ${\tilde{f}}:{\mathfrak{D}}^{m}\longrightarrow D$ satisfying that

1. (a)

${\tilde{f}}(|):=f(\#)$, 2. (b)

for $t=t^{l}\vee_{i}t^{r}\in{\mathcal{B}}_{n}^{m}$,

[TABLE]

As the elements of the set $\bigcup_{n\geq 1}{\mathcal{B}}_{n}^{m}$ do not contain any subtree of the form $(t\vee_{i}w)\vee_{j}u$, for $0\leq i\leq j\leq m$, we have that ${\tilde{f}}$ is well-defined. On the other hand, as $D$ is a $\mbox{\it Dyck}^{m}$ algebra, the way products $*_{i}$ are defined, implies that ${\tilde{f}}$ is a $\mbox{\it Dyck}^{m}$ algebra homomorphism. $\square$

For any vector space $V$, equation 1.2.1 shows that a basis of the free algebra ${\mathfrak{D}}^{m}(V)$ over $V$ is given by

[TABLE]

where $X$ is a basis of $V$.

1.3. The generating series associated to $\mbox{\it Dyck}^{m}$

Let

[TABLE]

be the Fuss-Catalan number, for $m,n\geq 1$.

The integer $d_{m,n}$ is the number of planar rooted $m$-ary trees, or equivalently the number of $m$-Dyck paths of length $n$, see for instance [6], [8], [37] or [21]. The generating series of $d_{m,n}$ is

[TABLE]

We want to see that the number of elements of ${\mathcal{B}}_{n}^{m}$ is $d_{m,n}$, for $m,n\geq 1$.

Notation 1.3.1**.**

Given a family of colored trees $t_{1},\dots,t_{p}$ and a family of integers $0\leq i_{1},\dots,i_{p}\leq m$, we denote by

1. (1)

$\Omega_{i_{1},\dots,i_{p}}^{L}(t_{1},\dots,t_{p})$ the colored tree

[TABLE] 2. (2)

$\Omega_{i_{1},\dots,i_{p}}^{R}(t_{1},\dots,t_{p})$ the colored tree

[TABLE]

That is

Remark 1.3.2**.**

For any tree $t\in{\mathcal{Y}}_{n-1}^{m}$ there exist unique non negative integers $p$ and $q$, such that

[TABLE]

for unique families of colored trees $t_{1},\dots,t_{p}$ and $w_{1},\dots,w_{q}$, and unique collections of integers $i_{1},\dots,i_{p}$ and $j_{1},\dots,j_{q}$ in $\{0,\dots,m\}$, with $i_{1}=j_{1}$.

In particular, $t=t^{l}\vee_{i_{1}}t^{r}$, for

[TABLE]

Notation 1.3.3**.**

Let $b_{m,n}$ denotes the number of elements of the set ${\mathcal{B}}_{n}^{m}$ introduced in Definition 1.2.4. We know that $b_{m,n}$ is the dimension of the subspace of homogeneous elements of degree $n$ of ${\mathfrak{D}}^{m}$. Let

[TABLE]

be the generating series of $\{b_{m,n}\}_{n\geq 1}$, for $m\geq 1$.

Lemma 1.3.4**.**

The generating series $f_{m}(x)$ is given by the following conditions

1. (1)

$b_{m,1}=1$, for $m\geq 1$, 2. (2)

$f_{m}(x)=x\cdot(1+f_{m}(x))^{m+1}$.

• Proof. For $n=1,2$, it is immediate to see that $b_{m,1}=1$ and $b_{m,2}=m$, for $m\geq 1$.

From Remark 1.3.2, we get that for any colored planar rooted tree $t\in{\mathcal{B}}_{n}^{m}$, there exist unique integers $p\geq 1$, $0\leq i_{1}<\dots<i_{p}\leq m$, and unique elements $t_{1},\dots,t_{p}$ in ${\mathcal{B}}_{m}$, such that

[TABLE]

For $n>2$, the coefficient $b_{m,n}$ of $x^{n}$ in $f_{m}(x)$ is

[TABLE]

On the other hand, we have that

[TABLE]

So, the coefficient of $x^{n}$ in $x(1+f_{m}(x))^{m+1}$ is

[TABLE]

which ends the proof. $\square$

Corollary 1.3.5**.**

For $m,n\geq 1$, the dimension of the subspace ${\mathfrak{D}}_{n}^{m}$ of homogeneous elements of degree $n$ of the free $\mbox{\it Dyck}^{m}$ algebra ${\mathfrak{D}}^{m}$ is equal to $d_{m,n}$.

The sequence of integers $\{d_{m,n}\}_{n\geq 1}$ describes different families of combinatorial objects, let us mention the planar rooted ${\mbox{}m+1}$-ary trees (see [44] and [24]) and the $m$-Dyck paths (see [6], [7]). In Section 4, we describe the free $\mbox{\it Dyck}^{m}$ algebra over one element on the space spanned by the set of all $m$-Dyck paths, for $m\geq 1$. The description of the operations $*_{i}$ on a Dyck are quite technical, but not difficult to explain in terms of the constructions developed in [6], [7] and [8]. Clearly, a similar construction may be done in terms of planar rooted $m$-ary trees. However, the last description is less natural, because it requires to cut the original tree in three or more subtrees and to glue them again in some complicated way.

Families of non-symmetric operads $\{{\mathcal{P}}^{m}\}_{m\geq 1}$, satisfying that the underlying subspace ${\mathcal{P}}_{n}^{m}$ has dimension $d_{m,n}$, for $n,m\geq 1$, have been defined previously in other authors. Let us mention

1. (1)

the operad ${}^{m}{\mathcal{P}}$ introduced by P. Leroux in [27]. The operad ${}^{2}{\mathcal{P}}$ coincides with the operad of dendriform algebras and the operad ${}^{3}{\mathcal{P}}$ has dimensions $d_{3,n}$, for $n\geq 1$. For $m\geq 3$, the dimension of ${}^{m}{\mathcal{P}}_{n}$ is different from $d_{m,n}$. 2. (2)

the operads of $m$-dendriform algebras, described by J.-C. Novelli in [35] 3. (3)

the operads ${\mbox{FCat}^{(m)}}$ introduced by S. Giraudo in [21].

All these operads, as well as $\mbox{\it Dyck}^{m}$, are equipped with an associative product, unique up to the product with some scalar.

Let us compare $\mbox{\it Dyck}^{2}$ with the operads ${}^{3}{\mathcal{P}}$ (defined in page 5 of [27]) and the operad of $2$-dendriform algebras (defined in [35], page 7). Novelli’s associative product $*$, as well as the associative product $*$ of ${}^{3}{\mathcal{P}}$ must be sent to $\lambda\cdot(*_{0}+*_{1}+*_{2})$, for some $\lambda\in{\mathbb{K}}$. So, we may assume that $*\mapsto*_{0}+*_{1}+*_{2}\in\mbox{\it Dyck}^{2}$.

Relations $(11)$, $(13)$ and $(16)$ of [35], completely determine the products $\succ$ and $\prec$ on a free $3$-dendriform algebra. Relations $(11)$ and $(16)$ coincide with the relation given in 1.1.2 for $i=2$ and $i=0$, respectively, while relation $(13)$ coincides with the relation 1.1.1, for $(i,j)=(0,2)$. So, the product $\succ$ maps to $*_{0}$, and the product $\prec$ maps to $*_{2}$.

The product $\circ$ of [35] must map to a linear combination of $*_{0},*_{1}$ and $*_{2}$. Replacing in relation $(12)$, we get that $\circ\mapsto a*_{0}+b*_{1}$. But, as $*$ maps to $*_{0}+*_{1}+*_{2}$, we get that $\circ$ maps to $*_{1}$. In this case, replacing $\succ$, $\circ$ and $\prec$ by $*_{0}$, $*_{1}$ and $*_{2}$, respectively, we see that equations $(14)$ and $(15)$ of [35], state that

1. (i)

$(u*_{2}v)*_{1}w=u*_{1}(v*_{1}w+v*_{0}w)$, 2. (ii)

$(u*_{1}v+u*_{0}v)=u*_{0}(v*_{1}w)$,

which are not valid in a free $\mbox{\it Dyck}^{2}$ algebra.

For $n>2$, the relations $(38)$, $(39)$, $(40)$ and $(42)$ of $m$-dendriform algebras, described in page $15$ of [35], imply that if the products $\succ,\circ_{1},\dots,\circ_{m-1}$ and $\succ$ map to linear combinations of $*_{0},\dots,*_{m}$, then there exists $0\leq k<m$ such that $\succ\mapsto*_{0}+\dots+*_{k}$, $\prec\mapsto*_{m}$ and $\circ_{i}$ maps to a linear combination of the products $*_{0},\dots,*_{m-1}$, for $1\leq i<m$. A straighforward calculation shows that in this case relation $(41)$ cannot hold in a $\mbox{\it Dyck}^{m}$ algebra.

Similar arguments hold when we compare the operad ${}^{3}{\mathcal{P}}$ with $\mbox{\it Dyck}^{2}$. The associative product $\succ+\prec$ maps to $*_{0}+*_{1}+*_{2}$. As Leroux’s products $\succ$ and $\prec$ are uniquely determined on free objects by relations $(1),(2)$ and $(3)$, we have two possibilities, to map $\succ$ to $*_{0}+*_{1}$ and $\prec$ to $*_{2}$, or to map $\succ$ to $*_{0}$ and $\prec$ to $*_{1}+*_{2}$. Both choices are equivalent, so we select the first option.

The product $\bullet$ must map to a linear combination of $*_{0}$, $*_{1}$ and $*_{2}$, but using equation $(6)$ we get that $\bullet\mapsto a*_{0}+b*_{1}$, where $a\neq b$. Applying it to equation $(4)$, we obtain

[TABLE]

which has no solution in a free $\mbox{\it Dyck}^{m}$ algebra.

Giraudo’s operad ${\mbox{FCat}^{(1)}}$ is spanned by products $\bullet_{0}$ and $\bullet_{1}$ which satisfy the relations

[TABLE]

, for $i,j\geq 0$ and $i+j\leq 1$.

The first equation is $(x\bullet_{0}y)\bullet_{0}z=x\bullet_{0}(y\bullet_{0}z)$, implies that $\bullet_{0}$ is associative. So, we we may assume that $\bullet_{0}\mapsto*_{0}+*_{1}$ in the operad $\mbox{\it Dyck}^{1}$ of dendriform algebras.

The third equation is $(x\bullet_{1}y)\bullet_{1}z=x\bullet_{1}(y\bullet_{0}z)$. If we look for a solution $\bullet_{1}=a*_{0}+b*_{1}$, we get that $a=0$ and $b=1$. So, the product $\bullet_{1}$ maps to $*_{1}$.

Replacing in the second equation $(x\bullet_{0}y)\bullet_{1}z=x\bullet_{0}(y\bullet_{1}z)$, we obtain that $(x*_{0}y+x*_{1}y)*_{1}z=x*_{0}(y*_{1}z)+x*_{1}(y*_{1}z)$. This equation implies that $(x*_{1}y)*_{1}z=x*_{1}(y*_{1}z)$, which is false in a free $\mbox{\it Dyck}^{1}$ algebra. So, the operad ${\mbox{FCat}^{(1)}}$ is not dendriform, and therefore ${\mbox{FCat}^{(1)}}\neq\mbox{\it Dyck}^{1}$. Applying analogous arguments, it is easily seen that ${\mbox{FCat}^{(m)}}\neq\mbox{\it Dyck}^{m}$, for $m\geq 1$.

2. A simplicial object in the category of non-symmetric algebraic operads.

2.1. The simplicial structure of $\{\mbox{\it Dyck}^{m}\}_{m\geq 0}$

Let ${\mbox{}\mbox{\it Dyck}^{m}{\mbox{-alg}}}$ denotes the category of $\mbox{\it Dyck}^{m}$ algebras over ${\mathbb{K}}$. We want to show that the family of categories $\{{\mbox{}\mbox{\it Dyck}^{m}{\mbox{-alg}}}\}_{m\geq 0}$, where ${\mbox{}\mbox{\it Dyck}^{0}{\mbox{-alg}}}$ is the category of associative algebras, is equipped with face and degeneracy functors, which define a co-simplicial object in the category of small categories.

For any pair of algebraic operads ${\mathcal{P}}$ and ${\mathcal{Q}}$, there exists a bijection between functors ${\mathbb{F}}:{\mbox{}{\mathcal{P}}{\mbox{-alg}}}\longrightarrow{\mbox{}{\mathcal{Q}}{\mbox{-alg}}}$ and operad morphisms from ${\mathcal{Q}}$ to ${\mathcal{P}}$. So, we get a simplicial object in the category of non-symmetric operads.

Definition 2.1.1**.**

Let $(D,*_{0},\dots,*_{m})$ be a $\mbox{\it Dyck}^{m}$ algebra.

1. (1)

Define the product ${\underline{*}}_{j}$, for $0\leq j\leq m+1$, by

[TABLE]

The binary operations $\{{\underline{*}}_{j}\}_{0\leq j\leq m+1}$ define a $\mbox{\it Dyck}^{m{\mbox{+}1}}$ algebra structure on the vector space $D$. We denote it by ${\mathbb{F}}_{i}(D)$. 2. (2)

For $0\leq i\leq m-1$, let ${\overline{*}}_{j}$ be the product on $D$ given by

[TABLE]

The vector space $D$, equipped with the products ${\overline{*}}_{i}$, for $0\leq i\leq m-1$, is a $\mbox{\it Dyck}^{m{\mbox{-}1}}$ algebra, which we denote ${\mathbb{S}}_{i}(D)$.

Remark 2.1.2**.**

For $m\geq 1$, the map $D\mapsto{\mathbb{F}}_{i}(D)$ defines a functor from the category of $\mbox{\it Dyck}^{m}$ algebras to the category of $\mbox{\it Dyck}^{m{\mbox{\it+}1}}$ algebras, for $0\leq i\leq m$. Similarly, the map $D\mapsto{\mathbb{S}}_{i}(D)$ gives a functor from the category of $\mbox{\it Dyck}^{m}$ algebras to the category of $\mbox{\it Dyck}^{m{\mbox{\it-}1}}$ algebras, for $0\leq i\leq m$.

Remark 2.1.2 implies that ${\mathbb{F}}_{i}:\mbox{\it Dyck}^{m{\mbox{\it+}1}}\longrightarrow\mbox{\it Dyck}^{m}$ and

${\mathbb{S}}_{i}:\mbox{\it Dyck}^{m}\longrightarrow\mbox{\it Dyck}^{m{\mbox{\it-}1}}$ are operad morphisms, for $0\leq i\leq m$.

The following result is immediate to verify.

Lemma 2.1.3**.**

Let NonSym be the category of non symmetric algebraic operads. The collection $\{\mbox{\it Dyck}^{m}\}_{m\geq 1}$, equipped with the operad morphisms ${\mathbb{F}}_{i}:\mbox{\it Dyck}^{m{\mbox{\it+}1}}\longrightarrow\mbox{\it Dyck}^{m}$ and ${\mathbb{S}}_{j}:\mbox{\it Dyck}^{m}\longrightarrow\mbox{\it Dyck}^{m{\mbox{\it+}1}}$, for $0\leq i\leq m+1$ and $0\leq j\leq m$, is a simplicial object in the category NonSym.

2.2. Functors ${\mathbb{S}}_{i}$ preserve free objects

For $0\leq i\leq m$, we want to show that the image under the functor ${\mathbb{S}}_{i}$ of a free $\mbox{\it Dyck}^{m}$ algebra is free as a $\mbox{\it Dyck}^{m{\mbox{\it-}1}}$ algebra. From Remark 1.2.1, it suffices to prove that the image ${\mathbb{S}}_{i}({\mathfrak{D}}^{m})$, of the free $\mbox{\it Dyck}^{m}$ algebra over one element, is free as a $\mbox{\it Dyck}^{m{\mbox{\it-}1}}$ algebra.

In order to do that, we need to introduce a new basis of the vector space ${\mathfrak{D}}^{m}$.

Definition 2.2.1**.**

Given $0\leq k\leq m$, define ${\mathcal{B}}_{n}^{m,k}$ to be the set of planar binary rooted trees with $n$ leaves, with the vertices colored by the elements of $\{0,\dots,m\}$ satisfying that, for any vertex $v$, the tree $t_{v}$ fulfills the following conditions

1. (i)

if $t_{v}=\Omega_{i_{1},\dots,i_{p}}^{L}(t_{1},\dots,t_{p})$, with $i_{1}\neq k$, then either $i_{2}=k$ or $i_{2}>i_{1}$, 2. (ii)

if $t_{v}=\Omega_{k,\dots,i_{p}}^{L}(t_{1},\dots,t_{p})=\Omega_{k,\dots,j_{q}}^{R}(w_{1},\dots,w_{q})$, then $i_{s}\in\{k+1,\dots,m\}$, for $2\leq s\leq p$, and $j_{h}\in\{0,\dots,k\}$, for $2\leq h\leq q$.

The basis ${\mathcal{B}}^{m}$, introduced in Definition 1.2.4 coincides with the set ${\mathcal{B}}^{m,m}$.

Note that the basis ${\mathcal{B}}^{m,k}$ is obtained from the set of all colored planar binary rooted trees, by replacing certain patterns, in the following way

1. (1)

for $0\leq i<j\leq m$ and $i\neq k$, we replace $(t_{1}\vee_{i}t_{2})\vee_{j}t_{3}$ by $t_{1}\vee_{i}(t_{2}\vee_{j}t_{3})$, 2. (2)

for $k<j\leq m$, we replace $t_{1}\vee_{k}(t_{2}\vee_{j}t_{3})$ by $(t_{1}\vee_{k}t_{2})\vee_{j}t_{3}$, 3. (3)

$(t_{1}\vee_{k}t_{2})\vee_{k}t_{3}$ is replaced by $\sum_{j=0}^{k}t_{1}\vee_{k}(t_{2}\vee_{j}t_{3})-\sum_{i=k+1}^{m}(t_{1}\vee_{i}t_{2})\vee_{k}t_{3}$.

We first prove that the set ${\mathcal{B}}^{m,k}$ is a basis of ${\mathfrak{D}}^{m}$, for $0\leq k<m$.

Proposition 2.2.2**.**

For $n\geq 1$ and $0\leq k\leq m$, there exists a bijective map $\Theta_{n}:{\mathcal{B}}_{n}^{m}\longrightarrow{\mathcal{B}}_{n}^{m,k}$ satisfying that

1. (1)

$\Theta_{n}(t)=t$, for any $t\in{\mathcal{B}}_{n}^{m}\cap{\mathcal{B}}_{n}^{m,k}$, 2. (2)

when the color of the root of $t$ is $k$, the color of the root of $\Theta_{n}(t)$ belongs to $\{k,\dots,m\}$, 3. (3)

when the color of the root of $t$ is different from $k$, the color of the root of $\Theta_{n}(t)$ is the same than the color of the root of $t$, 4. (4)

the elements $t$ and $\Theta_{n}(t)$ describe the same element in ${\mathfrak{D}}^{m}$.

• Proof. For $n=1$, the map $\Theta_{1}$ is the identity of ${\mathcal{B}}_{2}^{m}={\mathcal{B}}_{2}^{m,k}$.

For $n=2$, we have that $t\notin{\mathcal{B}}_{2}^{m}\cap{\mathcal{B}}_{2}^{m,k}$ if, and only if, $t=|\vee_{k}(|\vee_{i}|)$, with $k<i$. In this case, condition 1.1.1 implies that $t$ represents the same element than $(|\vee_{k}|)\vee_{i}|$, which belongs to ${\mathcal{B}}_{2}^{m,k}$. So, we define $\Theta(t):=(|\vee_{k}|)\vee_{i}|$.

For $n\geq 3$, note that

1. (i)

if $t=t^{l}\vee_{i}t^{r}\in{\mathcal{B}}_{n}^{m}$ with $i\neq k$, then the color of the root of $\Theta_{n_{\small 1}}(t^{l})$ is largest than $i$. So, the element $\Theta_{n_{\small 1}}(t^{l})\vee_{i}\Theta_{n_{\small 2}}(t^{r})\in{\mathcal{B}}_{n}^{m,k}$. Therefore, we get $\Theta_{n}(t):=\Theta_{n_{\small 1}}(t^{l})\vee_{i}\Theta_{n_{\small 2}}(t^{r})$, 2. (ii)

Suppose that the color of the root of a tree $t\in{\mathcal{B}}_{n}^{m}$ is $k$. If $\Theta_{n_{\small 2}}(t^{r})=\Omega_{i_{1},\dots,i_{p}}^{R}(w_{1},\dots,w_{p})$, with $i_{l}\leq k$ for $1\leq l\leq p$, then the color of the root of $t^{l}$ is largest than $k$. Therefore, the element $\Theta_{n_{\small 1}}(t^{l})\vee_{k}\Theta_{n_{\small 2}}(t^{r})$ belongs to ${\mathcal{B}}_{n}^{m,k}$ and represents the same element than $t$ in ${\mathfrak{D}}^{m}$. In this case, we get $\Theta_{n}(t):=\Theta_{n_{\small 1}}(t^{l})\vee_{k}\Theta_{n_{\small 2}}(t^{r})$.

The unique case which is more complicated is when there exists some $2\leq l\leq p$ such that $k<j_{l}$, where $t=\Omega_{k,j_{2},\dots,j_{p}}^{R}(t^{l},t_{2}^{r},\dots,t_{p}^{r})$.

Let $l_{0}$ be the smallest integer such that $k<j_{\small{l_{0}}}$. We have that $j_{l}\leq k$ for $2\leq l<l_{0}$, and that $t_{j_{\small{l_{0}}}}^{r}=\Omega_{s_{1},\dots,s_{q}}^{L}(w_{1},\dots,w_{q})$, with $j_{l_{0}}<s_{1}<\dots<s_{q}$ because $t_{j_{\small{l_{0}}}}^{r}\in{\mathcal{B}}^{m}$.

Using that $k<j_{l_{0}}<s_{1}<\dots<s_{k}$, and applying several times condition 1.1.1, we get that $t$ represents the same element that the tree

[TABLE]

where $w_{q+1}:=\Omega_{j_{2},\dots,j_{\small{l_{0}}-1}}^{R}(t_{2}^{r},\dots,t_{j_{\small l_{0}}-1}^{r})$ and $t^{l}=\Omega_{r_{1},\dots,r_{h}}^{L}(t_{1}^{l},\dots,t_{h}^{l})$.

As $t\in{\mathcal{B}}_{n}^{m}$, we obtain that $k<r_{1}<\dots<r_{h}$, and that $j_{l_{0}}<s_{1}<\dots<s_{q}$. Therefore, we get that

[TABLE]

The color of the root of the tree

[TABLE]

is $j_{l_{0}}>k$, which implies that the tree $\Omega_{j_{l_{0}},j_{l_{0}+1},\dots,j_{p}}^{R}(\Theta(t)^{l},\Theta(t_{j_{\small{l_{0}+1}}}^{r}),\dots,\Theta(t_{p}^{r}))$ represents the same element than $t$ in ${\mathfrak{D}}^{m}$. So, we define

[TABLE]

We need to show that $\Theta$ is bijective. For $n=1,2$ the result is immediate. For $n>2$, we apply a recursive argument on $n$.

Suppose that there exist $\Theta_{r}^{-1}:{\mathcal{B}}_{r}^{m,k}\longrightarrow{\mathcal{B}}_{r}^{m}$, for all $1\leq r<n$, satisfying that

1. (a)

the composition $\Theta_{r}\circ\Theta_{r}^{-1}$ is the identity on ${\mathcal{B}}_{r}^{m.k}$, 2. (b)

if the root of $t$ is colored by a element smaller or equal than $k$, then the root of $\Theta_{r}^{-1}(t)$ is colored by the same element. 3. (c)

if the root of $t$ is colored by a element $j$ with $k<j$, then the root of $\Theta_{r}^{-1}(t)$ is colored by an element in $\{k,j\}$.

Let $t=t^{l}\vee_{j}t^{r}\in{\mathcal{B}}_{r}^{m,k}$.

For $j<k$, the root of $t^{l}$ is colored by $h>j$, thus the root of $\Theta^{-1}(t^{l})$ is colored by an element largest than $j$.

We get that $\Theta^{-1}(t)=\Theta^{-1}(t^{l})\vee_{j}\Theta^{-1}(t^{r})$, and it is immediate to see that $\Theta\circ\Theta^{-1}(t)=t$.

For $j=k$, we have that $t^{l}=\Omega_{s_{1},\dots,s_{p}}^{L}(t_{1}^{l},\dots,t_{p}^{l})$ with $k<s_{l}$, for $1\leq l\leq p$. The root of $\Theta^{-1}(t^{l})$ is colored by an element largest than $k$, which implies that

[TABLE]

It is easily seen that $\Theta\circ\Theta^{-1}(t)=t$.

For $j>k$, if $t^{l}=\Omega_{s_{1},\dots,s_{p}}^{L}(t_{1}^{l},\dots,t_{p}^{l})$, for $j<s_{1}<\dots<s_{p}$ and $1\leq l\leq p$, then

[TABLE]

If $s_{h}=k$, for some $1\leq h\leq p$, then $h$ is unique. As $k<s_{h-1}$, by condition 1.1.1, the tree $\Omega_{s_{h-1},k,s_{h+1},\dots,s_{p}}^{L}(t_{h-1}^{l},t_{h}^{l},\dots,t_{p}^{l})$ represents the same element than the tree

[TABLE]

Since $k<j<s_{1}<\dots<s_{h-1}$, applying the same argument several times, we get that $t$ represents the same element than the tree

[TABLE]

where

[TABLE]

for $t_{h}^{l}=\Omega_{r_{1},\dots,r_{q}}^{L}(w_{1},\dots,w_{q})$.

So, $\Theta_{n}^{-1}(t):=\Omega_{k,s_{h+1},\dots,s_{p}}^{L}(\Theta^{-1}(w),\Theta^{-1}(t_{h+1}^{l}),\dots,\Theta^{-1}(t_{p}^{l}))$, and the definition of $\Theta$ shows that $\Theta_{n}\circ\Theta_{n}^{-1}(t)=t$, which ends the proof. $\square$

Corollary 2.2.3**.**

For any $0\leq k\leq m$, the set ${\mathcal{B}}^{m,k}={\displaystyle\bigcup_{n\geq 1}{\mathcal{B}}_{n}^{m,k}}$ is a basis of the underlying vector space of the free algebra ${\mathfrak{D}}^{m}$.

Notation 2.2.4**.**

Let $X$ be a set, we denote by ${\mathfrak{D}}^{m}(X)$ the free $\mbox{\it Dyck}^{m}$ algebra over $X$. The graded set $\bigcup_{n\geq 1}{\mathcal{B}}_{n}^{m,k}$ is denoted by ${\mathcal{B}}^{m,k}$.

Lemma 2.2.5**.**

For any integer $0\leq k<m$, the image of ${\mathfrak{D}}^{m}$ under the functor ${\mathbb{S}}_{k}$ is generated as $\mbox{\it Dyck}^{m{\mbox{-}1}}$ algebra by the graded set ${\mathcal{A}}^{m,k}$ of all colored trees $t$ in ${\mathcal{B}}^{m,k}$, satisfying that either $n=1$, or $n>1$ and the root of $t$ is colored by $k$.

• Proof. The $\mbox{\it Dyck}^{m{\mbox{-}1}}$ algebra structure of ${\mathfrak{D}}^{m}$ is given by the products ${\overline{*}}_{j}=\begin{cases}*_{j},&{\rm for}\ 0\leq j<k,\\ *_{k}+*_{k+1},&{\rm for}\ j=k,\\ *_{j-1},&{\rm for}\ k<j<m.\end{cases}$

The underlying vector space of ${\mathbb{S}}_{k}({\mathfrak{D}}^{m})$ is equal to ${\mathfrak{D}}^{m}$. As the set ${\mathcal{B}}^{m,k}$ is a basis of ${\mathfrak{D}}^{m}$ as a ${\mathbb{K}}$-vector space, it suffices to see that any element in ${\mathcal{B}}^{m,k}$ belongs to the $\mbox{\it Dyck}^{m{\mbox{-}1}}$ algebra generated by the set ${\mathcal{A}}^{m,k}$, under the operations ${\overline{*}}_{0},\dots,{\overline{*}}_{m-1}$.

We proceed by induction on the degree $n$. For $n=1,2$, the result is immediate.

For $t=t^{l}\vee_{i}t^{r}\in{\mathcal{B}}_{n}^{m,k}$, the recursive hypothesis states that the trees $t^{l}$ and $t^{r}$ are obtained by applying the products ${\overline{*}}_{0},\dots,{\overline{*}}_{m-1}$ to elements of the set ${\mathcal{A}}^{m,k}$ of degree smaller than $n$.

We have to analize three different cases

1. (1)

for $i<k$, we have that $t=t^{l}\ {\overline{*}}_{i}\ t^{r}$. As $t^{l}$ and $t^{r}$ are elements in the $\mbox{\it Dyck}^{m{\mbox{-}1}}$ algebra generated by ${\mathcal{A}}^{m,k}$, so is $t$, 2. (2)

for $i=k$, as $t\in{\mathcal{B}}_{n}^{m,k}$, we get that $t\in{\mathcal{A}}^{m,k}$, 3. (3)

for $i=k+1$, we have that $t=t^{l}\ {\overline{*}}_{k}\ t^{r}\ {-}\ t^{l}*_{k}t^{r}$ and the root of $t^{l}$ is colored by $j$, with $j>k+1$ or $j=k$.

As $t^{l}$ and $t^{r}$ belong to the free $\mbox{\it Dyck}^{m{\mbox{\it-}1}}$ algebra ${\mathfrak{D}}^{m{\mbox{\it-}1}}({\mathcal{A}}^{m,k})$, the tree $t^{l}\ {\overline{*}}_{k}\ t^{r}$ is in ${\mathfrak{D}}^{m{\mbox{\it-}1}}({\mathcal{A}}^{m,k})$.

On the other hand, either $t^{l}*_{k}t^{r}\in{\mathcal{A}}^{m,k}$, or

[TABLE]

for some colored tree $w=w^{l}\vee_{h}w^{r}$ and $h>k$.

Applying a recursive argument on the degrees of the elements $t^{l}\vee_{k}w^{l}$ and $w^{r}$ the result follows. 4. (4)

For $i>k+1$, we have that $t=t^{l}\vee_{i-1}t^{r}$ belongs to ${\mathfrak{D}}^{m{\mbox{\it-}1}}({\mathcal{A}}^{m,k})$ by recursive hypothesis.

$\square$

Lemma 2.2.5 states that, for any vector space $V$, ${\mathbb{S}}_{k}({\mathfrak{D}}^{m})$ is a quotient of the free $\mbox{\it Dyck}^{m{\mbox{\it-}1}}$ algebra ${\mathfrak{D}}^{m{\mbox{\it-}1}}({\mathcal{A}}^{m,k})$. For $X$ finite, the dimension of the subspace of homogeneous elements of degree $n$ in ${\mathbb{S}}_{k}(\mbox{\it Dyck}^{m}(X))$ is $d_{m,n}|X|^{n}$.

So, to prove that ${\mathbb{S}}_{k}({\mathfrak{D}}^{m}(X))$ is isomorphic to ${\mathfrak{D}}^{m{\mbox{\it-}1}}({\mathcal{A}}^{m,k}(X))$, it suffices to show that the dimension of the subspace of homogeneous elements of degree $n$ in ${\mathfrak{D}}^{m{\mbox{\it-}1}}({\mathcal{A}}^{m,k})$ is $d_{m,n}$, where ${\mathcal{A}}^{m,k}$ is the set of trees in ${\mathcal{B}}_{n}^{m}$ with the vertices colored by $\{0,\dots,m\}$ and the root colored by $k$.

Recall that, for any graded vector space $V=\bigoplus_{n\geq 1}V_{n}$ such that each $V_{n}$ is finite dimensional, the generating series of $V$ is $v(x):={\displaystyle\sum_{n\geq 1}{\mbox{dim}_{{\mathbb{K}}}(V_{n})}x^{n}}$.

Lemma 2.2.6**.**

Let $d_{m}(x)$ be the generating series of the free $\mbox{\it Dyck}^{m}$ algebra ${\mathfrak{D}}^{m}$. We have that

[TABLE]

for all $0\leq k\leq m$.

• Proof. Clearly, it suffices to prove the result for $k=m-1$. Let $g_{m}(x)$ be the inverse series of $d_{m}(x)$ ($g$ exists because $d(0)=0$).

Since $x\cdot(1+d_{m}(x))^{m+1}=d_{m}(x)$, replacing $x$ by $g_{m}(x)$ we obtain that

[TABLE]

which implies that $(1+x)\cdot g_{m}(x)=g_{m-1}(x)$. So, replacing $x$ by $d_{m}(x)$ and applying $d_{m-1}(x)$ to both sides, we get the desired formula

[TABLE]

$\square$

Applying Lemmas 2.2.5 and 2.2.6, we get the following result.

Proposition 2.2.7**.**

The $\mbox{\it Dyck}^{m{\mbox{\it-}1}}$ algebra ${\mathbb{S}}_{k}({\mathfrak{D}}^{m}(X))$ is free, for any $0\leq k\leq m-1$.

• Proof. Applying Lemmas 2.2.5 and 2.2.6, it suffices to prove that the number of elements in ${\mathcal{A}}_{n}^{m,k}$ is $d_{m,n-1}$, for $0\leq k\leq m$.

The number of elements of ${\mathcal{B}}_{n-1}^{m,k}$ is $d_{m,n-1}$. To end the proof we define a bijective map $\theta_{n}$ from ${\mathcal{B}}_{n-1}^{m,k}$ to ${\mathcal{A}}_{n}^{m,k}$, for $n\geq 2$.

For $n=2$, $\theta_{1}(|)$ is the unique planar binary rooted tree with two leaves and the root colored by $k$.

Let $t=t^{l}\vee_{h}t^{r}$ be an element of ${\mathcal{B}}_{n-1}^{m,k}$.

1. (1)

For $h>k$, let $t=\Omega_{h,i_{2},\dots,i_{p}}^{L}(t^{r},t_{2},\dots,t_{p})$.

1. (a)

If $i_{p}>\dots>i_{2}>h>k$, then $\theta_{n}(t):=t\vee_{k}|.$ 2. (b)

If there exists one integer $1\leq s\leq p$ such that $i_{s}=k$, then the $s$ is unique and $\theta_{n}(t)$ is defined by the formula

[TABLE] 2. (2)

For $h\leq k$, let $t=\Omega_{h,j_{2},\dots,j_{q}}^{R}(t^{l},w_{2},\dots,w_{q})$.

1. (a)

If $j_{i}\leq k$ for any $2\leq i\leq q$, then $\theta_{n}(t):=|\vee_{k}t$. 2. (b)

Otherwise, let $2\leq s\leq q$ be the minimal integer satisfying that $j_{s}>k$. As $t\in{\mathcal{B}}^{m,k}$, we know that $k\notin\{h,j_{1},\dots,j_{s-1}\}$. Define $\theta_{n}(t)$ as the element

[TABLE]

It is not difficult to verify that $\theta_{n}$ is bijective for all $n\geq 2$. So, the result is proved. $\square$

The following result is a straightforward consequence of Proposition 2.2.7.

Theorem 2.2.8**.**

The image of a free $\mbox{\it Dyck}^{m}$ algebra ${\mathfrak{D}}^{m}(X)$ under the functor ${\mathbb{S}}_{k}$ is a free $\mbox{\it Dyck}^{m{\mbox{-}1}}$ algebra, for $0\leq k<m$.

Note that, by composing the functors ${\mathbb{S}}_{k}$, we get that the associative algebra $({\mathfrak{D}}^{m}(X),*_{0}+\dots+*_{m})$ is free, for any set $X$.

3. Dendriform posets

Let ${\mathcal{Y}}:=\{{\mathcal{Y}}_{n},\leq_{Ta}\}_{n\geq 1}$ be the family of sets ${\mathcal{Y}}_{n}$ of planar binary rooted trees. The vector space $V_{\small{\mathcal{Y}}}:=\bigoplus_{n\geq 1}{\mathbb{K}}[{\mathcal{Y}}_{n}]$ admits a structure of dendriform algebra, satisfying that the products $\succ$, $\prec$ and $*=\succ+\prec$ are defined in terms of intervals of the Tamari order. This result was described in [30]. In the same work, an analogous result was proved for the family ${\mathbb{\Sigma}}=\{\Sigma_{n}\}_{n\geq 1}$, where $\Sigma_{n}$ is the set of permutations of $n$ elements, where the Tamari order is replaced by the weak Bruhat order (see also [17]). Both results were extended in [38] to the families of all surjective maps and of all planar rooted trees, where the partial orders of these sets are generalizations of the weak Bruhat order and of the Tamari order, respectively.

We describe the conditions that a family of partially ordered sets ${\mathbb{P}}=\{P_{n}\}_{n\geq 1}$ must fulfill in order to

1. (1)

get a natural structure of dendriform algebra on the vector space $V_{\mathbb{P}}=\bigoplus_{n\geq 1}{\mathbb{K}}[P_{n}]$, spanned by the graded set $\bigcup_{n\geq 1}P_{n}$. 2. (2)

get an $\mbox{\it Dyck}^{m}$ algebra structure on the vector space spanned by the set $\bigcup_{n\geq 1}{\mbox{Simp}(P_{n})^{m}}$ of $m$-simpleces of the partially ordered sets $P_{n}$, $n\geq 1$.

This type of families of partially ordered sets provide examples of $\mbox{\it Dyck}^{m}$ algebras.

3.1. Basic constructions

Definition 3.1.1**.**

A dendriform poset is a family of partially orederd sets ${\mathbb{P}}=\{P_{n}\}_{n\geq 1}$, equipped with four binary graded products $/$, $\perp$, $\top$ and $\backslash$, satisfying the following conditions

1. (1)

for $n,r\geq 1$, the maps $/,\perp,\top,\backslash:P_{n}\times P_{r}\longrightarrow P_{n+m}$, preserve the orders (where the partial order of $P_{n}\times P_{m}$ is the componentwise one), 2. (2)

for any pair of elements $x\in P_{n}$ and $y\in P_{m}$, the interval $[x/y;x\backslash y]$ is the disjoint union of the intervals $[x/y;x\perp y]$ and $[x\top y;x\backslash y]$, 3. (3)

for any elements $x\in P_{n}$, $y\in P_{r}$ and $z\in P_{s}$, there exists a bijective order preserving map $\varphi_{x,y,z}$ from the set

[TABLE]

to the set

[TABLE]

satisfying that

1. (a)

the restriction of $\varphi_{x,y,z}$ to the set

[TABLE]

gives a bijection with the set

[TABLE] 2. (b)

the restriction of $\varphi_{x,y,z}$ to the set

[TABLE]

gives a bijection with the set

[TABLE]

, 4. (4)

if there exist $u\leq v$ in $P_{n+r}$ satisfying that $x/y\leq u\leq x\backslash y$ and $z/w\leq v\leq z\backslash w$, for some elements $x,z\in P_{n}$ and $y,w\in P_{r}$, then $x\leq z$ in $P_{n}$ and $v\leq w$ in $P_{r}$. 5. (5)

if there exist $u\leq v$ in $P_{n+r}$ satisfying that $x/y\leq u\leq x\perp y$ and $x\top y\leq v\leq x\backslash y$, for some elements $x\in P_{n}$ and $y\in P_{r}$, then $v\not\leq u$.

Notation 3.1.2**.**

Let ${\mathbb{P}}=\{P_{n}\}_{n\geq 1}$ be a dendriform poset. We denote by $V_{\small{\mathbb{P}}}:=\bigoplus_{n\geq 1}{\mathbb{K}}[P_{n}]$ the graded vector space spanned by ${\mathbb{P}}$.

Definition 3.1.3**.**

Let ${\mathbb{P}}=\{P_{n}\}_{n\geq 1}$ be a dendriform poset. The products $\succ$ and $\prec$ are defined by

[TABLE]

for any elements $x,y\in\bigcup_{n\geq 1}P_{n}$. Both products are extended by linearity to the vector space $V_{\small{\mathbb{P}}}$.

Proposition 3.1.4**.**

Let ${\mathbb{P}}=\{P_{n}\}_{n\geq 1}$ be a dendriform poset. The vector space $V_{\small{\mathbb{P}}}$ equipped with the products $\succ$ and $\prec$ is a dendriform (or $\mbox{\it Dyck}^{1}$) algebra. Conversely, if $(V_{\small{\mathbb{P}}},\succ,\prec)$ is a dendriform algebra, then ${\mathbb{P}}$ satisfies conditions $(2)$ and $(3)$ of Definition 3.1.1.

• Proof. Condition $(2)$ is equivalent to $x*y=x\succ y+x\prec y=\sum_{x/y\leq u\leq x\backslash y}u$, for any elements $x,y\in V_{\small{\mathbb{P}}}$.

Condition $(3)$ is equivalent to

1. (1)

$x*(y*z)=(x*y)*z$. 2. (2)

$x\succ(y\succ z)=(x*y)\succ z$, 3. (3)

$x\prec(y*z)=(x\prec y)\prec z$,

for any elements $x,y,z\in\bigcup_{n\geq 1}P_{n}$, therefore for any $x,y$ and $z$ in $V_{\small{\mathbb{P}}}$.

As $*=\succ+\prec$ is associative, we get that

[TABLE]

for any elements $x,y$ and $z$ in $V_{\small{\mathbb{P}}}$.

So, $(V_{\small{\mathbb{P}}},\succ,\prec)$ is a dendriform algebra. $\square$

3.2. Examples

1) The facial order on surjective maps

Notation 3.2.1**.**

Let ${\mbox{Surj}_{n}^{r}}$ denotes the set of surjective maps from $[n]$ to $[r]$, for $1\leq r\leq n$. For any $f\in{\mbox{Surj}_{n}^{r}}$, we denote it by the tuple of its images $f=(f(1),\dots,f(n))$. Note that ${\mbox{Surj}_{n}^{n}}$ is the group $\Sigma_{n}$ of permutations of $n$ elements, for $n\geq 1$.

We denote the disjoint union $\bigcup_{r=1}^{n}{\mbox{Surj}_{n}^{r}}$ by ${\mbox{Surj}_{n}}$.

For $1\leq i\leq{n-1}$, let $s_{i}$ be the element of $\Sigma_{n}$ which exchanges $i$ and $i+1$. The surjective map $\tau_{i}\in{\mbox{Surj}_{n}^{n{\mbox{-}1}}}$ is the map given by

[TABLE]

Let $S$ and $T$ be two disjoint subsets of $[n]$, we say that $S<T$ if $s<t$, for any pair of elements $s\in S$ and $t\in T$.

For any $S\subseteq[n]$ and any $f\in{\mbox{Surj}_{n}}$, let $f|_{S}$ denote the restriction of $f$ to the set $S$. Suppose that $S=\{j_{1},\dots,j_{s}\}$ and that $r$ is a positive integer, we denote by $S+r$ the set $\{j_{1}+r,\dots,j_{s}+r\}$.

Definition 3.2.2**.**

Let $f:[n]\longrightarrow{\mathbb{N}}$ be a map, the standardization of $f$ is the unique surjective map ${\mbox{std}(f)}:[n]\longrightarrow[|{\mbox{Im}(f)}|]$ satisfying that ${\mbox{std}(f)}(i)<{\mbox{std}(f)}(j)$ if, and only if, $f(i)<f(j)$, for all elements $1\leq i,j\leq n$.

Definition 3.2.3**.**

The facial order on the set ${\mbox{Surj}_{n}}$ is the transitive relation $\leq_{fa}$ spanned by the following covering relations

1. (1)

If $f^{-1}(i)<f^{-1}(i+1)$, then $f\lessdot\tau_{i}\circ f$, for $1\leq i<|{\mbox{Im}(f)}|$, 2. (2)

if $f^{-1}(i)=\{j_{1}<\dots<j_{s}\}$, then $f\lessdot f_{k}$, where $f_{k}$ is the map

[TABLE]

for any $1\leq k<s$.

In order to define a dendriform poset structure on Surj, we must define the graded products $/$, $\perp$, $\top$ and $\backslash$, these operations were introduced in [38].

Definition 3.2.4**.**

Let $f\in{\mbox{Surj}_{n}^{s}}$ and $g\in{\mbox{Surj}_{r}^{h}}$ be two surjective maps, define

[TABLE]

Proposition 3.2.5**.**

The family ${\mbox{\bf Surj}}=(\{{\mbox{Surj}_{n}}\}_{n\geq 1},\leq_{fa})$ is a dendriform poset.

• Proof. A recursive argument on $n\geq 1$ shows that the products $/,\perp,\top$ and $\backslash:{\mbox{Surj}_{n}}\times{\mbox{Surj}_{r}}\longrightarrow{\mbox{Surj}_{n+r}}$ preserve the order.

The proof that $V_{\small{\mbox{\bf Surj}}}$ is a dendriform algebra with the products $f\succ g:=\sum_{f/g\leq_{fa}u\leq_{fa}f\perp g}u$ and $f\prec g:=\sum_{f\top g\leq_{fa}u\leq_{fa}f\backslash g}u$, and that the associative product $*:=\succ+\prec$ is given by $f*g:=\sum_{f/g\leq_{fa}u\leq_{fa}f\backslash g}u$ was done in [38], we refer to this reference for the details of the proof. This result implies that Surj fulfills the second and third conditions of Definition 3.1.1. We need to prove that it also satisfies the last two ones.

Note that, for any pair of elements $f\in{\mbox{Surj}_{n}}$ and $g\in{\mbox{Surj}_{r}}$, we have that ${\mbox{std}((f\odot g)|_{\{1,\dots,n\}}}=f$ and ${\mbox{std}((f\odot g)|_{\{n+1,\dots,n+r\}}}=g$, where $\odot$ is any element of the set $\{/,\perp,\top,\backslash\}$.

On the other hand, for $f\leq_{fa}g$ in ${\mbox{Surj}_{n}^{s}}$ and any subset $S\subseteq[r]$, it is immediate to see that $f|_{S}\leq_{fa}g|_{S}$.

If $u\leq_{fa}w$ in ${\mbox{Surj}_{n+r}}$ and there exist $f_{1},f_{2}\in{\mbox{Surj}_{n}}$ and $g_{1},g_{2}\in{\mbox{Surj}_{r}}$ such that

[TABLE]

then $f_{1}={\mbox{std}(u|_{\{1,\dots,n\}})}\leq_{fa}{\mbox{std}(w|_{\{1,\dots,n\}})}=f_{2}$ and $g_{1}={\mbox{std}(u|_{\{n+1,\dots,n+r\}})}\leq_{fa}{\mbox{std}(w|_{\{n+1,\dots,n+r\}})}=g_{2}$. Therefore, $f_{1}\leq_{fa}f_{2}$ and $g_{1}\leq_{fa}g_{2}$, and the fourth condition is satisfied.

Let $f_{1},f_{2}\in{\mbox{Surj}_{n}}$ and $g_{1},g_{2}\in{\mbox{Surj}_{r}}$, and let $u$ and $w$ be elements in ${\mbox{Surj}_{n+r}}$ satisfying that

[TABLE]

We need to prove that $w\not\leq_{fa}u$.

Suppose that $u\in{\mbox{Surj}_{n+r}^{k}}$ and $w\in{\mbox{Surj}_{n+r}^{h}}$ satisfy that $w\leq_{fa}u$. The definition of $\leq_{fa}$ implies that, for any $i\in w^{-1}(k)$ there exists $1\leq j_{i}\leq i$ such that $u(j_{i})=k$.

On the other hand, suppose that $f_{i}\in{\mbox{Surj}_{n}^{s_{i}}}$ and $g_{i}\in{\mbox{Surj}_{r}^{h_{i}}}$, for $i=1,2$. We have that

1. (a)

$f_{1}/g_{1}$ and $f_{1}\perp g_{1}$ belong to ${\mbox{Surj}_{n+r}^{s_{1}+h_{1}}}$ satisfy

[TABLE]

which implies that $u^{-1}(k)=g_{1}^{-1}(h_{1})+n$. 2. (b)

$f_{2}\top g_{2}$ and $f_{2}\backslash g_{2}$ belong to ${\mbox{Surj}_{n+r}^{s_{2}+h_{2}}}$, and satisfy

1. (i)

$(f_{2}\top g_{2})^{-1}(s_{2}+h_{2})=f_{2}^{-1}(s_{2})\bigcup(g_{2}^{-1}(h_{2})+n),$ 2. (ii)

$(f_{2}\backslash g_{2})^{-1}(s_{2}+h_{2})=f_{2}^{-1}(s_{2}),$

which imply that

[TABLE]

Point $(a)$ implies that $u^{-1}(k)\subseteq\{n+1,\dots,n+r\}$. Let $i\in f_{2}^{-1}(s_{2})$, by $(b)$ we know that $w(i)=l$. So, there exists $1\leq j_{i}\leq i\leq n$ such that $u(j_{i})=k$, which is false. So, $w\not\leq_{fa}u$, and the last condition of Definition 3.1.1 is fulfilled. $\square$

Remark 3.2.6**.**

The symmetric group $\Sigma_{n}$ is generated by the permutations $s_{i}$, for $1\leq i\leq{\mbox{}n-1}$. The length of a permutation $\sigma\in\Sigma_{n}$ is the minimal integer $l(\sigma)$ satisfying that $\sigma=s_{j_{1}}\cdot\dots\cdot s_{j_{l(\sigma)}}$. The left weak Bruhat order $\leq_{B}$ on $\Sigma_{n}$ is the transitive relation spanned by

[TABLE]

The restriction of the order $\leq_{fa}$ to the subset $\Sigma_{n}={\mbox{Surj}_{n}^{n}}$ gives the left weak Bruhat order.

Proposition 3.2.13 and Remark 3.2.6 imply that the family ${\mathbb{\Sigma}}=\{\Sigma_{n}\}_{n\geq 1}$ is also a dendriform poset, via the inclusion map $\Sigma_{n}\subseteq{\mbox{Surj}_{n}}$. It suffices to note that ${\mathbb{\Sigma}}$ is closed under the products $/$. $\perp$ and $\backslash$, and that the product $\top$ is given by

[TABLE]

2) The partial order on planar rooted trees

Let ${\mathcal{T}}_{n}^{r}$ be the set of planar rooted trees with $n+1$ leaves and $n-r$ internal vertices, for $0\leq r\leq n-1$. Note that ${\mathcal{T}}_{n}^{0}={\mathcal{Y}}_{n}$ is the set of planar binary rooted trees with $n+1$ leaves.

Notation 3.2.7**.**

We denote by ${\mathcal{T}}_{n}$ the set ${\displaystyle\bigcup_{r=0}^{n-1}{\mathcal{T}}_{n}^{r}}$. For a collection of planar rooted trees $t^{1},\dots,t^{p}$, $\bigvee(t^{1},\dots,t^{p})$ denotes the planar tree obtained by joining the roots of the trees $t^{1},\dots,t^{p}$, disposed from left to right, to a new root. If $t^{i}\in{\mathcal{T}}_{n_{i}}^{r_{i}}$, for $1\leq i\leq p$, then $\bigvee(t^{1},\dots,t^{r})\in{\mathcal{T}}_{n}^{r}$, where $n+1=n_{1}+\dots+n_{p}+p$ and $r+2=r_{1}+\dots+r_{p}+p$.

For $t\in{\mathcal{T}}_{n}$, we denote $|t|=n$.

There exist surjective maps (see [36], [38]) $\Gamma_{n}:{\mbox{Surj}_{n}}\longrightarrow{\mathcal{T}}_{n}$, satisfying that the inverse image $\Gamma_{n}^{-1}(t)$ is an interval in the facial order of ${\mbox{Surj}_{n}}$, for $n\geq 1$. The facial order $\leq_{fa}$ of the set ${\mbox{Surj}_{n}}$ induces a partial order $\leq_{T}$ on the set ${\mathcal{T}}_{n}$, for $n\geq 1$.

These partially ordered sets were studied in [38], in order to show that the dendriform (and tridendriform) algebra structure, defined in [31] on the graded vector space ${\mathbb{K}}[{\mathcal{T}}_{\infty}]:=\bigoplus_{n\geq 1}{\mathbb{K}}[{\mathcal{T}}_{n}]$, may be described in terms of intervals of these orders. We briefly describe the main results needed to show that the family $(\{{\mathcal{T}}_{n}\}_{n\geq 1},\leq_{T})$ is a dendriform poset, for a more detailed description we refer to [38].

The following result was proved in [38].

Lemma 3.2.8**.**

The facial order of ${\mbox{Surj}_{n}}$ induces a partial order $\leq_{T}$ on ${\mathcal{T}}_{n}$, which extends the Tamari order defined on the set ${\mathcal{Y}}_{n}$ of planar binary rooted trees. The partial order $\leq_{T}$ is completely determined by the following conditions

1. (1)

for $n=2$, the order is described by

2. (2)

for a family of trees $t^{1},\dots,t^{p}$, suppose that $t^{i}=\bigvee(t^{i1},\dots,t^{is})$, with $s>1$, for some $1\leq l<p$. In this case

[TABLE] 3. (3)

for a family of trees $t^{1},\dots,t^{p}$, with $p\geq 3$,

[TABLE]

for any pair $0<i<j<p$. 4. (4)

given two families of trees $t^{1},\dots,t^{p}$ and $w^{1},\dots,w^{p}$, with $p\geq 2$, satisfying that $t^{l}\leq_{T}w^{l}$, for $1\leq l\leq p$,

[TABLE]

Definition 3.2.9**.**

For any tree $t\in{\mathcal{T}}_{n}$ and any $0\leq l\leq n$, the $i^{th}$* restriction* of $t$ is the pair $(t_{(1)}^{l},t_{(2)}^{l})\in{\mathcal{T}}_{l}\times{\mathcal{T}}_{n-l}$ recursively defined by

1. (a)

for $i=0$, $(t_{(1)}^{0},t_{(2)}^{0}):=(|,t)$, 2. (b)

for $i=n$, $(t_{(1)}^{n},t_{(2)}^{n}):=(t,|)$, 3. (c)

for $n=2$ and $i=1$, $(t_{(1)}^{1},t_{(2)}^{1}):=(\bigvee(|,|),\bigvee(|,|))$, for any $t\in{\mathcal{T}}_{2}$, 4. (d)

For $t=\bigvee(t^{1},\dots,t^{p})$, with $|t^{j}|=n_{j}$, and $1<l+1<n_{1}+\dots+n_{p}+p$, define

[TABLE]

where $1\leq k\leq p$ is the unique integer such that $n_{1}+\dots+n_{k-1}+k\leq l+1<n_{1}+\dots+n_{k}+k+1$.

In fact the $i^{th}$ restriction of a tree $t$ is the result of slicing in two the tree $t$ following the path from the $i^{th}$ leaf to the root. Moreover, a long but simple recursive argument shows the following result.

Lemma 3.2.10**.**

For any tree pair of trees $t\leq_{T}w$ in ${\mathcal{T}}_{n}$ and any $0\leq j\leq n$, we have that $t_{(i)}^{j}\leq_{T}w_{(i)}^{j}$, for $i=1,2$.

Definition 3.2.11**.**

Let $t\in{\mathcal{T}}_{n}$ and $w\in{\mathcal{T}}_{r}$ be two planar rooted trees. The planar rooted trees $t/w$ and $t\backslash w$ in ${\mathcal{T}}_{n+r}$ are defined as follows

1. (1)

$t/w$ is the tree obtained by grafting the root of $t$ to the leftmost leaf of $w$, 2. (2)

$t\backslash w$ is the tree obtained by grafting the root of $w$ to the rightmost leaf of $t$.

Remark 3.2.12**.**

For $t\in{\mathcal{T}}_{n}$ and $w\in{\mathcal{T}}_{r}$, we have that $((t/w)_{(1)}^{n},(t/w)_{(2)}^{n})=(t,w)=((t\backslash w)_{(1)}^{n},(t\backslash w_{(2)}^{n}))$.

Proposition 3.2.13**.**

The family ${\mathbb{\mathcal{T}}}:=(\{{\mathcal{T}}_{n}\}_{n\geq 1},\leq_{T})$ is a dendriform poset.

• Proof. In [31] three binary operations $\succ$, $\cdot$ and $\prec$ are defined on the vector space ${\mathbb{K}}[{\mathcal{T}}_{\infty}]$, spanned by the set of all planar rooted trees, in such a way that the products $*_{0}:=\succ$ and $*_{1}:=\cdot+\prec$ define a dendriform (or $\mbox{\it Dyck}^{1}$) algebra.

For any pair of planar rooted trees $t=\bigvee(t^{1},\dots,t^{p})$ and $w=\bigvee(w^{1},\dots,w^{q})$, define the trees

[TABLE]

where $/$ and $\backslash$ are the products described in Definition 3.2.11. The maps $/,\perp,\top$ and $\backslash$ are morphisms of partially ordered sets from ${\mathcal{T}}_{n}\times{\mathcal{T}}_{r}$ to ${\mathcal{T}}_{n+r}$, for $n,r\geq 1$. Moreover, the products $*_{0}$ and $*_{1}$ are given by the equalities

[TABLE]

for any planar rooted trees $t$ and $w$. We refer to [38] for the details of the proof.

The previous result implies that the family ${\mathbb{\mathcal{T}}}$, satisfies the first three conditions of Definition 3.1.1.

Suppose that $t_{1}/w_{1}\leq_{T}u\leq_{T}t_{1}\backslash w_{1}$ and $t_{2}/w_{2}\leq_{T}v\leq_{T}t_{2}\backslash w_{2}$ are such that $u\leq_{T}v$, for elements $t_{1},t_{2}\in{\mathcal{T}}_{n}$ and $w_{1},w_{2}\in{\mathcal{T}}_{r}$. Applying Lemma 3.2.10 and Remark 3.2.12, we get that

[TABLE]

which implies that condition $(4)$ of Definition 3.1.1 is satisfied.

To end the proof we need to see that if $t_{1}/w_{1}\leq_{T}u\leq_{T}t_{1}\perp w_{1}$ and $t_{2}\top w_{2}\leq_{T}v\leq_{T}t_{2}\backslash w_{2}$, for two pairs of elements $t_{1},t_{2}\in{\mathcal{T}}_{n}$ and $w_{1},w_{2}\in{\mathcal{T}}_{r}$, then $v\not\leq_{T}u$.

Suppose that $t_{i}=\bigvee(t_{i}^{1},\dots,t_{i}^{p_{i}})$ and $w_{i}=\bigvee(w_{i}^{1},\dots,w_{i}^{q_{i}})$, for $i=1,2$, and that $u=\bigvee(u^{1},\dots,u^{s})$ and $v=\bigvee(v^{1},\dots,v^{k})$.

From Definition 3.2.8, it is immediate to verify that $v\leq_{T}u$ implies that $|v^{1}|\leq|u^{1}|$.

On the other hand, $t_{1}/w_{1}\leq_{T}u\leq_{T}t_{1}\perp w_{1}$ implies that $l=|t_{1}|+|w_{1}^{1}|$. But the condition $t_{2}/w_{2}\leq_{T}v\leq_{T}t_{2}\backslash w_{2}$ implies that $|v^{1}|=|t_{1}^{1}<|t_{1}|\leq|u^{1}|$. So, we get that $v\not\leq_{T}u$, and the proof is over. $\square$

As in the first example, the restriction of the facial order on planar rooted trees to binary planar rooted trees, gives the Tamari order $\leq_{Ta}$ on the set ${\mathcal{Y}}_{n}$, for $n\geq 1$ (see [25], and [41]). Using the results of [30], it is easy to see that the family ${\mathbb{\mathcal{Y}}}:=\{{\mathcal{Y}}_{n}\}_{n\geq 1}$, equipped with $\leq_{Ta}$, is also a dendriform poset.

3.3. The $\mbox{\it Dyck}^{m}$ algebra of $m$-simplexes

Let ${\mathbb{P}}=\{P_{n}\}_{n\geq 1}$ be a dendriform poset.

Notation 3.3.1**.**

For a positive integer $n$, the ${\mbox{Simp}({\mathbb{P}})}_{n}^{m}$ be the set of all $m$-simplexes of the partially ordered set $P_{n}$. That is,

[TABLE]

For $m$ fixed, let ${\mbox{Ord}^{m}}({\mathbb{P}}):=\bigoplus_{n\geq 1}{\mathbb{K}}[{\mbox{Simp}({\mathbb{P}})}_{n}^{m}]$ be the graded vector space spanned by the set $\displaystyle{\bigcup_{n\geq 1}{\mbox{Simp}({\mathbb{P}})}_{n}^{m}}$ of all $m$-simplexes, for $n\geq 1$.

We know that the space $V_{\small{\mathbb{P}}}$ has a natural structure of dendriform algebra. The aim of the present subsection is to describe a structure of $\mbox{\it Dyck}^{m}$-algebra on ${\mbox{Ord}^{m}}({\mathbb{P}})$.

Definition 3.3.2**.**

Let ${\mathbb{P}}$ be a dendriform poset. For a pair of elements ${\underline{x}}=(x_{1},\dots,x_{m})\in{\mbox{Simp}({\mathbb{P}})}_{n}^{m}$ and ${\underline{y}}=(y_{1},\dots,y_{m})\in{\mbox{Simp}({\mathbb{P}})}_{r}^{m}$,, and any $0\leq i\leq m$, let ${\mathcal{I}}^{i}({\underline{x}},{\underline{y}})$ be the set of all elements ${\underline{u}}\in{\mbox{Simp}({\mathbb{P}})}_{n+r}^{m}$ satisfying that

1. (i)

$x_{j}/y_{j}\leq u_{j}\leq x_{j}\perp y_{j}$, for $1\leq j\leq{\mbox{\it m-i}}$ 2. (ii)

$x_{j}\top y_{j}\leq u_{j}\leq x_{j}\backslash y_{j}$, for ${\mbox{\it m-i}}<j\leq m$.

In order to define products $*_{i}$ on the vector space ${\mbox{Ord}^{m}}({\mathbb{P}})$, for $0\leq i\leq m$, we need to prove some results.

Lemma 3.3.3**.**

Let $0\leq i<j\leq m$. For any pair of elements $({\underline{x}},{\underline{y}})\in{\mbox{Simp}({\mathbb{P}})}^{m}$, we have that the set ${\mathcal{I}}^{i}({\underline{x}},{\underline{y}})$ is equal to the set of all elements ${\underline{u}}\in{\mbox{Simp}({\mathbb{P}})}_{n+r}^{m}$ satisfying that

1. (i)

$x_{k}/y_{k}\leq u_{k}\leq x_{k}\backslash y_{k}$, for $1\leq k\leq m-j$, 2. (ii)

$x_{k}/y_{k}\leq u_{k}\leq x_{k}\perp y_{k}$, for $m-j<k\leq m-i$, 3. (iii)

$x_{j}\top y_{j}\leq u_{j}\leq x_{j}\backslash y_{j}$, for $m-i<k\leq m$.

• Proof. As ${\mathbb{P}}$ is a dendriform poset, the interval $[x/y;x\backslash y]$ is equal to the disjoint union of $[x/y;x\perp y]$ and $[x\top y;x\backslash y]$, for any elements $x,y\in\bigcup_{n\geq 1}P_{n}$.

So, any ${\underline{u}}\in{\mathcal{I}}^{i}({\underline{x}},{\underline{y}})$ satisfies that $x_{k}/y_{k}\leq u_{k}\leq x_{k}\backslash y_{k}$, for $1\leq k\leq m-j$, which proves the inclusion of ${\mathcal{I}}^{i}({\underline{x}},{\underline{y}})$.

We need to see that any ${\underline{u}}\in{\mbox{Simp}({\mathbb{P}})}_{n+r}^{m}$, such that

1. (i)

$x_{k}/y_{k}\leq u_{k}\leq x_{k}\backslash y_{k}$, for $1\leq k\leq m-j$ 2. (ii)

$x_{k}/y_{k}\leq u_{k}\leq x_{k}\perp y_{k}$, for $m-j<k\leq m-i$,

satisfies that $x_{k}/y_{k}\leq u_{k}\leq x_{k}\perp y_{k}$, for $1\leq k\leq m-i$.

As $i<j$, we get that $m-j<m-i$. Suppose that for some $1\leq k\leq m-j$, we have that $x_{k}\top y_{k}\leq u_{k}\leq x_{k}\backslash y_{k}$.

In this case $u_{k}\leq u_{m-i}$ and $x_{m-i}/y_{m-i}\leq u_{m-i}\leq x_{m-i}\perp y_{m-i}$, which contradicts condition $(5)$ of Definition 3.1.1.

So, $x_{k}/y_{k}\leq u_{k}\leq x_{k}\perp y_{k}$, for all $1\leq k\leq m-j$, which implies that ${\underline{u}}\in{\mathcal{I}}^{i}({\underline{x}},{\underline{y}})$. $\square$

The following Lemma is a consequence of similar arguments than the ones applied in the proof of Lemma 3.3.3.

Lemma 3.3.4**.**

Let $0\leq j\leq i-1<m$. For any pair of elements $({\underline{x}},{\underline{y}})\in{\mbox{Simp}({\mathbb{P}})}^{m}$, we have that the set ${\mathcal{I}}^{i}({\underline{x}},{\underline{y}})$ is equal to the set of all elements ${\underline{u}}\in{\mbox{Simp}({\mathbb{P}})}_{n+r}^{m}$ satisfying that

1. (i)

$x_{k}/y_{k}\leq u_{k}\leq x_{k}\backslash y_{k}$, for $1\leq k\leq m-i$, 2. (ii)

$x_{k}/y_{k}\leq u_{k}\leq x_{k}\backslash y_{k}$, for $m-i<k\leq m-j$, 3. (iii)

$x_{j}\top y_{j}\leq u_{j}\leq x_{j}\backslash y_{j}$, for $m-j<k\leq m$.

Definition 3.3.5**.**

The product $*_{i}:{\mbox{Ord}^{m}}({\mathbb{P}})\otimes{\mbox{Ord}^{m}}({\mathbb{P}})\longrightarrow{\mbox{Ord}^{m}}({\mathbb{P}})$, for $0\leq i\leq m$, is defined by

[TABLE]

for any pair of elements ${\underline{x}}$ and ${\underline{y}}$ in ${\mbox{Simp}({\mathbb{P}})}^{m}$. The products are extended by linearity to the vector space ${\mbox{Ord}^{m}}({\mathbb{P}})$.

Theorem 3.3.6**.**

For any dendriform poset ${\mathbb{P}}$ and any $m\geq 1$, the vector space ${\mbox{Ord}^{m}}({\mathbb{P}})$, equipped with the products $*_{i}$, for $0\leq i\leq m$, is a $\mbox{\it Dyck}^{m}$ algebra.

• Proof. We need to prove that $({\mbox{Ord}^{m}}({\mathbb{P}}),*_{0},\dots,*_{m})$ satisfies the relations of Definition 1.1.3. For $n=1$, the result was proved in Proposition 3.1.4.

For $n\geq 2$, we have that From Definition 3.1.1, we have that

[TABLE]

for any $1\leq k\leq m$ and ${\underline{x}},{\underline{y}},{\underline{z}}\in{\mbox{Simp}({\mathbb{P}})}^{m}$.

In all the equalities above, from condition $(4)$ of Definition 3.1.1, we have that if $v_{1}\leq\dots\leq v_{m}$, then $u_{1}\leq\dots\leq u_{m}$ and $w_{1}\leq\dots\leq w_{m}$.

1. (1)

Let $0\leq i<j\leq m$. We have that

[TABLE]

where the sum is taken over all the elements ${\underline{v}}\in{\mbox{Simp}({\mathbb{P}})}^{m}$ such that there exists ${\underline{u}}\in{\mbox{Simp}({\mathbb{P}})}^{m}$ satisfying that

1. (a)

$y_{k}/z_{k}\leq u_{k}\leq y_{k}\perp z_{k}$ and $x_{k}/u_{k}\leq v_{k}\leq x_{k}\perp u_{k}$, for $1\leq k\leq m-j$, 2. (b)

$y_{k}\top z_{k}\leq u_{k}\leq y_{k}\backslash z_{k}$ and $x_{k}/u_{k}\leq v_{k}\leq x_{k}\perp u_{k}$, for $m-j<k\leq m-i$, 3. (c)

$y_{k}\top z_{k}\leq u_{k}\leq y_{k}\backslash z_{k}$ and $x_{k}\top u_{k}\leq v_{k}\leq x_{k}\backslash u_{k}$, for $m-i<k\leq m$.

As $m-j<m-i$ and $u_{1}\leq\dots\leq u_{m}$, Lemma 3.3.4 implies that we may change condition$(c)$ in the last paragraph, for the following condition

$(c^{\prime})$ for $m-i<k\leq m$, $y_{k}/z_{k}\leq u_{k}\leq y_{k}\backslash z_{k}$ and $x_{k}\top u_{k}\leq v_{k}\leq x_{k}\backslash u_{k}$.

Applying conditions $(1.1)$, $(1.2)$ and $(1.3)$ above, we get that

[TABLE]

where the sum is taken over all the elements ${\underline{v}}\in{\mbox{Simp}({\mathbb{P}})}^{m}$ such that there exists ${\underline{w}}\in{\mbox{Simp}({\mathbb{P}})}^{m}$ satisfying that

1. (a)

$x_{k}/y_{k}\leq w_{k}\leq x_{k}\backslash y_{k}$ and $w_{k}/z_{k}\leq v_{k}\leq w_{k}\perp z_{k}$, for $1\leq k\leq m-j$, 2. (b)

$x_{k}/y_{k}\leq w_{k}\leq x_{k}\perp y_{k}$ and $w_{k}\top z_{k}\leq s_{k}\leq w_{k}\backslash z_{k}$, for $m-j<k\leq m-i$, 3. (c)

$x_{k}\top y_{k}\leq w_{k}\leq x_{k}\backslash y_{k}$ and $w_{k}\top z_{k}\leq v_{k}\leq w_{k}\backslash z_{k}$, for $m-i<k\leq m$.

Again, as $m-j<m-i$ and $w_{1}\leq\dots\leq w_{m}$, Lemma 3.3.3 implies that condition $(a)$ may be changed into

$(a^{\prime})$ $x_{k}/y_{k}\leq w_{k}\leq x_{k}\perp y_{k}$ and $w_{k}/z_{k}\leq v_{k}\leq w_{k}\perp z_{k}$, for $1\leq k\leq m-j$,

which implies that ${\underline{x}}*_{i}({\underline{y}}*_{j}{\underline{z}})=({\underline{x}}*_{i}{\underline{y}})*_{j}{\underline{z}}$. 2. (2)

Let $0\leq i\leq m$, it is easy to show that

1. (a)

$\sum_{j=0}^{i}{\underline{y}}*_{j}{\underline{z}}=\sum_{u_{1}\leq\dots\leq u_{m}}{\underline{u}}$, where the sum is taken over all the elements ${\underline{u}}$ satisfying that

1. (i)

$y_{k}/z_{k}\leq u_{k}\leq y_{k}\perp z_{k}$, for $1\leq k\leq m-i$, 2. (ii)

$y_{k}/z_{k}\leq u_{k}\leq y_{k}\backslash z_{k}$, for $m-i<k\leq m$, 2. (b)

$\sum_{j=i}^{m}{\underline{x}}*_{j}{\underline{y}}=\sum_{w_{1}\leq\dots\leq w_{m}}{\underline{w}}$, where the sum is taken over all the elements ${\underline{w}}$ satisfying that

1. (i)

$x_{k}/y_{k}\leq w_{k}\leq x_{k}\backslash y_{k}$, for $1\leq k\leq m-i$, 2. (ii)

$x_{k}\top y_{k}\leq w_{k}\leq x_{k}\backslash y_{k}$, for $m-i<k\leq m$,

for elements ${\underline{x}},{\underline{y}}$ and ${\underline{z}}$ in ${\mbox{Simp}({\mathbb{P}})}^{m}$.

We get that

[TABLE]

where the sum is taken over all elements ${\underline{v}}\in{\mbox{Simp}({\mathbb{P}})}^{m}$ such that there exists ${\underline{u}}$ satisfying that

1. (a)

$y_{k}/z_{k}\leq u_{k}\leq y_{k}\perp z_{k}$ and $x_{k}/u_{k}\leq v_{k}\leq x_{k}\perp u_{k}$, for $1\leq k\leq m-i$, 2. (b)

$y_{k}/z_{k}\leq u_{k}\leq y_{k}\backslash z_{k}$ and $x_{k}\top u_{k}\leq v_{k}\leq x_{k}\backslash u_{k}$, for $m-i<k\leq m$.

Note that $v_{1}\leq\dots\leq v_{m}$, implies that ${\underline{u}}\in{\mbox{Simp}({\mathbb{P}})}^{m}$ from condition $(4)$ of Definition 3.1.1.

Applying conditions $(1.1)$, $(1.2)$ and $(1.3)$, we obtain

[TABLE]

where

1. (i)

$x_{k}/y_{k}\leq w_{k}\leq x_{k}\backslash y_{k}$ and $w_{k}/z_{k}\leq v_{k}\leq w_{k}\perp z_{k}$, for $1\leq k\leq m-i$, 2. (ii)

$x_{k}\top y_{k}\leq w_{k}\leq x_{k}\backslash y_{k}$ and $w_{k}\top z_{k}\leq v_{k}\leq w_{k}\backslash z_{k}$, for $m-i<k\leq m$,

for some ${\underline{w}}\in{\mbox{Simp}({\mathbb{P}})}^{m}$.

As

[TABLE]

with

• –

$x_{k}/y_{k}\leq w_{k}\leq x_{k}\backslash y_{k}$, for $1\leq k\leq m-i$,

• –

$x_{k}\top y_{k}\leq w_{k}\leq x_{k}\backslash y_{k}$, for $m-i<k\leq m$.

We get that

[TABLE]

which ends the proof.

$\square$

4. $\mbox{\it Dyck}^{m}$ algebras and $m$-Dyck paths

The dimension of the subspace of homogeneous elements of degree $n$ of the free $\mbox{\it Dyck}^{m}$ algebra ${\mathfrak{D}}^{m}$ is the number of $m$-Dyck paths of size $n$. We define a $\mbox{\it Dyck}^{m}$ algebra structure (isomorphic to ${\mathfrak{D}}^{m}$) on the vector space spanned by the set of all $m$-Dyck paths, and prove that this structure may be described in terms of the $m$-Tamari lattice introduced by F. Bergeron in [6].

4.1. $m$-Dyck paths

We describe basic notions bout $m$-Dyck paths. For more detailed constructions and the proofs of the results we refer to [7], [8] and [9].

Definition 4.1.1**.**

For $m,n\geq 1$, an $m$-Dyck path of size $n$ is a path on the real plan ${\mathbb{R}}^{2}$, starting at $(0,0)$ and ending at $(2nm,0)$, consisting of up steps $(m,m)$ and down steps $(1,-1)$, which never goes below the $x$-axis. Note that the initial and terminal points of each step lean on ${\mathbb{Z}}_{+}^{2}$.

Notation 4.1.2**.**

We denote by $\mbox{\it Dy}_{n}^{m}$ the set of all $m$-Dyck paths of size $n$. Define $\mbox{\it Dy}_{0}^{m}:=\{\bullet\}$, for $m\geq 1$.

For any $m\geq 0$, we denote by $\rho_{m}\in\mbox{\it Dy}_{1}^{m}$ the unique $m$-Dyck path of size one.

In order to define constructions on Dyck paths, we use the notation employed by M. Bousquet-Mélou, E. Fusy and the second author in [8].

Notation 4.1.3**.**

Let $P$ be an $m$-Dyck path. We denote by $\mathcal{UP}(P)$ the set of up steps of $P$, and by $\mathcal{DW}(P)$ the set of down steps of $P$.

Definition 4.1.4**.**

Let $u\in\mathcal{UP}(P)$ be an up step of an $m$-Dyck path $P$, the rank of $u$ is $k$ if $u$ is the $k^{\rm th}$ up step of $P$, counting from left to right.

A down step $d$ is at level $k$ if the last up step $u$ preceding $d$ has rank $k$.

As any up step $u$ in a Dyck path $P$ is determined by its rank, from now on we identify them, and denote the set of up steps of a Dyck path of size $n$ as $\mathcal{UP}(P)=\{1,\dots,n\}$.

Example 4.1.5**.**

The down steps of the following $2$-Dyck path are colored with their levels

Notation 4.1.6**.**

For a path $P\in\mbox{\it Dy}_{n}^{m}$ and an integer $1\leq k\leq n$, we denote by $\mathcal{DW}_{k}(P)$ the set of down steps of level $k$ of $P$ and by $L_{k}(P)$ the number of elements of $\mathcal{DW}_{k}(P)$. Note that $\mathcal{DW}_{k}(P)$ may be the empty set, and in this case $L_{k}(P)=0$. When no confusion is possible, we shall denote the last term of the sequence $L_{n}(P)$ simply by $L(P)$.

Note that $0\leq\L_{1}(P)+\dots+L_{j}(P)\leq mj$, for $1\leq j\leq n$. As a Dyck path $P\in\mbox{\it Dy}_{n}^{m}$ is uniquely determined by the sequence $L_{1}(P),\dots,L_{n-1}(P),L(P)$, we denote $P=((L_{1}(P),\dots,L_{(}P)))$.

For any integer $1\leq k\leq n$ such that $\mathcal{DW}_{k}(P)\neq\emptyset$, we identify the set $\mathcal{DW}_{k}(P)$ with the sequence of its down steps $\mathcal{DW}_{k}(P)=d_{1}^{k}d_{2}^{k}\dots d_{L_{k}(P)}^{k}$, ordered from left to right, and from top to bottom.

4.2. Basic operations on Dyck paths

We want to describe basic operations on Dyck paths that we need in the sequel.

Definition 4.2.1**.**

Let $P$ and $Q$ be two $m$-Dyck paths of sizes $n$ and $r$, respectively. For $0\leq i\leq L(P)$, define the $i^{th}$-concatenation of $P$ and $Q$, denoted $P\times_{i}Q$, as the Dyck path of size $n+r$ given by

[TABLE]

Definition 4.2.2**.**

An $m$-Dyck path $P$ is called prime if there does not exist a pair of $m$-Dyck paths $Q$ and $R$ such that $P=Q\times_{0}R$, with $P\neq Q$ and $P\neq R$.

Remark 4.2.3**.**

For any $m$-Dyck path of size $n$ there exist a unique composition $(n_{1},\dots,n_{r})$ of $n$ (with $n_{i}\geq 1$ for each $i$) and a unique family of prime Dyck paths $P_{1}\in\mbox{\it Dy}_{n_{1}}^{m},\ \dots,P_{r}\in\mbox{\it Dy}_{n_{r}}^{m}$ such that $P=P_{1}\times_{0}\dots\times_{0}P_{r}$.

The proof of the following Lemma is immediate.

Lemma 4.2.4**.**

Let $P\in\mbox{\it Dy}_{n_{1}}^{m}$ be a prime Dyck path and let $Q\in\mbox{\it Dy}_{n_{2}}^{m}$ be another Dyck path. For any $1\leq j\leq L(P)$, the Dyck path $P\times_{j}Q$ is prime.

Definition 4.2.5**.**

Let $P$ be an $m$-Dyck path of size $n$. The standard coloring of $P$ is a map $\alpha_{P}$ from the set of down steps $\mathcal{DW}(P)$ to the set $\{1,\dots,n\}$, described recursively as follows

1. (1)

For $P=\rho_{m}\in\mbox{\it Dy}_{1}^{m}$, $\alpha_{\rho_{m}}$ is the constant function $1$. 2. (2)

For $P=\bigvee_{d}(P_{0},\dots,P_{m})$, with $P_{j}\in\mbox{\it Dy}_{n_{j}}^{m}$, the set of down steps of $P$ is the disjoint union

[TABLE]

where the first subset $\{1,\dots,m\}$ corresponds to the down steps of $\rho_{m}$.

The map $\alpha_{P}$ is defined by

[TABLE]

where $0\leq j\leq m$.

For instance, the standard coloring of the $2$-Dyck path $P=((0,1,4,0,5,0,1,5))$ is

Remark 4.2.6**.**

Let $P$ be an $m$-Dyck path of size $n$.

1. (1)

We have that $|\alpha_{P}^{-1}(i)|=m$, for any $1\leq i\leq n$. 2. (2)

Let $\mathcal{DW}^{k}(P)=d_{1}^{k},d_{2},\dots,d_{L_{k}(P)}^{k}$ be the ordered sequence of down steps of level $k$ of $P$, for $1\leq k\leq n$.

The word $\omega_{k}^{P}=\alpha_{P}(d_{1}^{k})\dots\alpha_{P}(d_{L_{k}(P)}^{k})$ is decreasing for the usual order of the natural numbers. Moreover, the first $m$ digits of $\omega_{n}^{P}$ are $n$’s. 3. (3)

If $Q$ is another $m$-Dyck path, then $\mathcal{DW}(P\times_{i}Q)=\mathcal{DW}(P)\bigcup\mathcal{DW}(Q)$, and $\alpha_{P\times_{i}Q}$ is described by

[TABLE]

for any $0\leq i\leq L(P)$.

4.3. The $\mbox{\it Dyck}^{m}$ algebra structure on $m$-Dyck paths

Definition 4.3.1**.**

For any positive integer $n$, a weak composition of $n$ with $r+1$ parts is an ordered collection of non-negative integers ${\underline{\lambda}}=(\lambda_{0},\dots,\lambda_{r})$ such that $\lambda_{0}+\dots+\lambda_{r}=n$. We say that the length of ${\underline{\lambda}}$ is $r+1$.

Notation 4.3.2**.**

The set of all weak compositions of $n$ with $r+1$ parts is denoted by $\Lambda_{r}^{n}$. Given an $m$-Dyck path $P$ of size $n$ we denote the set of all weak compositions of $L(P)$ of length $r+1$ by $\Lambda_{r}(P)$.

Let $Q\in\mbox{\it Dy}_{s}^{m}$ be an $m$-Dyck path, such that $Q=Q_{1}\times_{0}\dots\times_{0}Q_{r}$ with $Q_{j}\in\mbox{\it Dyck}^{m}$ prime, for $1\leq j\leq r$. Given another Dyck path $P\in\mbox{\it Dy}_{n}^{m}$, suppose that ${\underline{\lambda}}=(\lambda_{0},\dots,\lambda_{r})$ is a weak composition of $L(P)$. Define a Dyck path $P*_{{\underline{\lambda}}}Q$ of size $n+s$ by the formula

[TABLE]

The product $*_{{\underline{\lambda}}}$ just divides the ordered set $\mathcal{DW}_{n}(P)$ of down steps of level $n$ of $P$ and glue, in order, the $i^{th}$ piece at the end of the path $Q_{i}$. If $\lambda_{0}>0$, the first $\lambda_{0}$ steps of $\mathcal{DW}_{n}(P)$ remain at the end of $P$.

Example 4.3.3**.**

Let $P=((2,3,1,6))\in\mbox{\it Dy}_{4}^{3}$ and let $Q=((1,4,4,3,2,3,4))$ be a $3$-Dyck path of size $7$, where we denote the Dyck paths following Notation 4.1.6. We have that that $Q=((1,4,4))\times_{0}\rho_{3}\times_{0}((2,3,4))$.

The word on the top level of $P$ is $\omega_{4}^{P}:=444331$. Consider the weak composition ${\underline{\lambda}}=(1,2,2,1)$ of $L(P)=6$ of length $4$. The path $P\times_{(1,2,2,1)}Q$ is the following

The last point of Remark 4.2.6 implies that for any $P\in\mbox{\it Dy}_{n}^{m}$, any $Q=Q_{1}\times_{0}\dots\times_{0}Q_{r}$, with $Q_{i}$ prime for $1\leq i\leq r$, and any ${\underline{\lambda}}\in\Lambda_{r}(P)$, the set of down steps of $P*_{{\underline{\lambda}}}Q$ is

[TABLE]

and the standard coloring $\alpha_{P*_{{\underline{\lambda}}}Q}$ is described by

[TABLE]

Notation 4.3.4**.**

Let $P$ be a Dyck path of size $n$, with $\mathcal{DW}_{n}(P)=d_{1}^{n},\dots,d_{L(P)}^{n}$, and let ${\underline{\lambda}}=(\lambda_{0},\dots,\lambda_{r})$ be a weak composition of $L(P)$. For $0\leq i\leq m$, we denote by $\Lambda_{r}^{i}(P)$ the set of all weak compositions ${\underline{\lambda}}$ of length $r+1$ such that the restriction $\alpha_{P}(d_{L(P){-}\lambda_{r}+1}^{n}),\dots,\alpha_{P}(d_{L(P)}^{n})$ of the word $\omega_{n}^{P}$ to its last $\lambda_{r}$ letters satisfies the following conditions:

1. (1)

any digit in the word $\alpha_{P}(d_{L(P)-\lambda_{r}+1}^{n}),\dots,\alpha_{P}(d_{L(P)}^{n})$ appears at most $i$ times, 2. (2)

there exists at least one integer $1\leq i_{0}\leq n$ such that $i_{0}$ appears exactly $i$ times in $\alpha_{P}(d_{L(P){-}\lambda_{r}+1}^{n}),\dots,\alpha_{P}(d_{L(P)}^{n})$.

For example, for $P=((0,2,1,3,4))\in\mbox{\it Dyck}c_{5}^{2}$, we get that $\omega_{5}^{P}=5,5,1,1$. So, ${\underline{\lambda}}_{1}=(1,1,2)$ belongs to $\Lambda_{2}^{2}(P)$, while ${\underline{\lambda}}_{2}=(0,3,1)$ belongs to $\Lambda_{2}^{1}(P)$.

Observe that

[TABLE]

The set of all weak compositions of $L(P)$ is the disjoint union {\displaystyle\bigcup_{r\geq 0}\bigl{(}\bigcup_{i=0}^{m}\Lambda_{r}^{i}(P)\bigr{)}}, for any $m$-Dyck path $P$ of size $n$.

The following result is a straightforward consequence of Lemma 4.2.4 and the definition of $*_{\lambda}$.

Lemma 4.3.5**.**

Let $P=P_{1}\times_{0}\dots\times_{0}P_{s}$ in $\mbox{\it Dy}_{n}^{m}$ and $Q=Q_{1}\times_{0}\dots\times_{0}Q_{r}$ in $\mbox{\it Dy}_{q}^{m}$ be two Dyck paths, where $P_{1},\dots,P_{s},Q_{1},\dots,Q_{r}$ are prime Dyck paths, and let ${\underline{\lambda}}\in\Lambda_{r}^{i}(P)$ be a weak composition. We have that

1. (1)

if $i>0$, then

[TABLE]

where $P_{s}*_{{\underline{\lambda}}}Q$ is prime. 2. (2)

if $i=0$, then ${\underline{\lambda}}=(\lambda_{0},\dots,\lambda_{r-1},0)$ and

[TABLE]

where $j_{0}$ is the maximal element of $\{0,\dots,r{-}1\}$ such that $\lambda_{j_{0}}\neq 0$.

The product on the graded vector space ${\mathbb{K}}[\mbox{\it Dy}^{m}]$, spanned by the set of all $m$-Dyck paths, is defined as follows.

Definition 4.3.6**.**

Let $P\in\mbox{\it Dy}_{n}^{m}$ and $Q\in\mbox{\it Dy}_{s}^{m}$ be two Dyck paths, such that $Q=Q_{1}\times_{0}\dots\times_{0}Q_{r}$ with $Q_{i}$ prime, $1\leq i\leq r$. For any integer $0\leq j\leq m$, define

[TABLE]

The product extends in a unique way to a linear map from ${\mathbb{K}}[\mbox{\it Dy}^{m}]\otimes{\mathbb{K}}[\mbox{\it Dy}^{m}]$ to ${\mathbb{K}}[\mbox{\it Dy}^{m}]$.

Example 4.3.7**.**

Consider the $2$-Dyck paths $P=((1,3))\in\mbox{\it Dy}_{2}^{2}$ and $Q=((0,2,4,2))=((0,2,4))\times_{0}\rho_{2}\in\mbox{\it Dy}_{4}^{2}$. Computing the products $P*_{0}Q$ and $P*_{1}Q$ , we get that

[TABLE]

Proposition 4.3.8**.**

Let $P\in\mbox{\it Dy}_{n}^{m}$ and $Q=Q_{1}\times_{0}\dots\times_{0}Q_{r}\in\mbox{\it Dy}_{p}^{m}$ be two Dyck paths, with $Q_{j}\in\mbox{\it Dy}_{p_{j}}^{m}$ prime for $1\leq j\leq r$.

1. (1)

For nonnegative integers $s\geq 1$ and $0\leq i<j\leq m$, the map

[TABLE]

which sends $({\underline{\lambda}},{\underline{\tau}})\mapsto({\underline{\lambda}},{\underline{\delta}}:=(\tau_{0},\dots,\tau_{s-1},\tau_{s}+\lambda_{r}))$ is bijective. 2. (2)

For any integer $0\leq i\leq m$, the map $\psi_{i}^{1}(P,Q)({\underline{\lambda}},{\underline{\tau}}):=$

[TABLE]

defines a bijection from the set $\{({\underline{\lambda}},{\underline{\tau}})\mid{\underline{\tau}}\in\Lambda_{s}^{0}(Q)\ {\rm and}\ {\underline{\lambda}}\in\Lambda_{r+s{-}j_{\underline{\tau}}}^{i}(P)\}$ to the set

[TABLE]

where $j_{\underline{\tau}}$ is the maximal integer $0\leq j\leq s-1$ such that $\tau_{j}>0$, and $\bigcup$ denotes the disjoint union. 3. (3)

For any integer $0\leq i\leq m$, the map

[TABLE]

which maps $({\underline{\lambda}},{\underline{\tau}})\mapsto({\underline{\lambda}},{\underline{\delta}}:=(\tau_{0},\dots,\tau_{s-1},\tau_{s}+\lambda_{r})$, is bijective.

• Proof. $(1)$ For the first point, let ${\underline{\lambda}}\in\Lambda_{r}^{i}(P)$ and ${\underline{\tau}}\in\Lambda_{s}^{j}(Q)$ be two weak compositions.

If $\mathcal{DW}_{n}(P)=d_{1}^{n},\dots,d_{L(P)}^{n}$ and $\mathcal{DW}_{p}(Q)=d_{1}^{p},\dots,d_{L(Q)}^{p}$, then

[TABLE]

The map $\psi_{ij}$ is defined by the formula

[TABLE]

Clearly, ${\underline{\lambda}}$ belongs to $\Lambda_{r}^{i}(P)$. On the other hand,

[TABLE]

which implies that the subset of the last $\tau_{s}+\lambda_{r}$ down steps of $P*_{{\underline{\lambda}}}Q$ is $d_{L(Q){-}\tau_{s}+1}^{p},\dots,d_{L(Q)}^{p},d_{L(P){-}\lambda_{r}+1}^{n},\dots,d_{L(P)}^{n}$.

Note that

1. (a)

$\alpha_{P*_{{\underline{\lambda}}}Q}(d_{L(P){-}\lambda_{r}+1}^{n})\dots\alpha_{P*_{{\underline{\lambda}}}Q}(d_{L(P)}^{n})$ is a sequence of elements in the set $\{1,\dots n\}$ such that any digit appears at most $i$ times. 2. (b)

$\alpha_{P*_{{\underline{\lambda}}}Q}(d_{L(Q){-}\tau_{s}+1}^{p})\dots\alpha_{P*_{{\underline{\lambda}}}Q}(d_{L(Q)}^{p})$ is a sequence of elements in the set $\{n+1,\dots,n+p\}$ where there exists at least one digit that appears $j$ times, and no digit appears more than $j$ times.

So, ${\underline{\delta}}$ belongs to $\Lambda_{s}^{j}(P*_{{\underline{\lambda}}}Q)$.

For any pair of weak compositions ${\underline{\lambda}}\in\Lambda_{r}^{i}(P)$ and ${\underline{\delta}}\in\Lambda_{s}^{j}(P*_{{\underline{\lambda}}}Q)$, we get that $\omega_{n+p}^{P*_{{\underline{\lambda}}}Q}$ is equal to

[TABLE]

As the expression $\alpha_{P}(d_{L(P){-}\lambda_{r}+1}^{n})\dots\alpha_{P}(d_{L(P)}^{n})$ is a word in the alphabet $\{1,\dots,n\}$ such that no digit appears more than $i$ times, and $i<j$, then ${\underline{\tau}}:=(\delta_{0},\dots,\delta_{s}{-}\lambda_{r})$ must belong to $\Lambda_{s}^{j}(Q)$.

It is immediate to prove that the map $({\underline{\lambda}},{\underline{\delta}})\mapsto({\underline{\lambda}},{\underline{\tau}})$ is inverse to $\psi_{ij}(P,Q)$, which ends the proof of $(1)$.

$(2)$ If ${\underline{\lambda}}\in\Lambda_{r+s{-}j_{\underline{\tau}}}^{i}(P)$ and ${\underline{\tau}}\in\Lambda_{s}^{0}(Q)$, then it is immediate to verify that

1. (i)

${\underline{\gamma}}=(\lambda_{0},\dots,\lambda_{r-1},\lambda_{r}+\dots+\lambda_{r+s{-}j_{\underline{\tau}}})$ belongs to $\Lambda_{r}^{j}(P)$, for $i\leq j\leq m$, 2. (ii)

${\underline{\delta}}=(\tau_{0},\dots,\tau_{j_{\underline{\tau}}}+\lambda_{r},\lambda_{r+1},\dots,\lambda_{r+s{-}j_{\underline{\tau}}})$ belongs to $\Lambda_{s}^{i}(P*_{\underline{\gamma}}Q)$, 3. (iii)

$\delta_{s}=\lambda_{r+s{-}j_{\underline{\tau}}}\leq\gamma_{r}=\lambda_{r}+\dots+\lambda_{r+s{-}j_{\underline{\tau}}}$.

Assume that we have two weak compositions ${\underline{\gamma}}=(\gamma_{0},\dots,\gamma_{r})\in{\displaystyle\bigcup_{j=i}^{m}\Lambda_{r}^{j}(P)}$ and ${\underline{\delta}}=(\delta_{0},\dots,\delta_{s})\in\Lambda_{s}^{i}(P*_{\underline{\gamma}}Q)$, such that $\delta_{s}\leq\gamma_{r}$.

Let $j_{0}$ be the maximal integer $0\leq j_{0}\leq s{-}1$, such that $\delta_{j_{0}}+\dots+\delta_{s}>\gamma_{r}$.

Define

1. (a)

${\underline{\lambda}}:=(\gamma_{0},\dots,\gamma_{r-1},\gamma_{r}{-}\delta_{j_{0}+1}{-}\dots-\delta_{s},\delta_{j_{0}+1},\dots,\delta_{s})$, 2. (b)

${\underline{\tau}}:=(\delta_{0},\dots,\delta_{j_{0}{-}1},\delta_{j_{0}}+\dots+\delta_{s}-\gamma_{r},0,\dots,0)$.

It is clear that ${\underline{\lambda}}\in\Lambda_{r+s-j_{0}}^{i}(P)$, ${\underline{\tau}}\in\Lambda_{s}^{0}(Q)$ and $\psi_{i}^{1}(P,Q)({\underline{\lambda}},{\underline{\tau}})=({\underline{\gamma}},{\underline{\delta}})$, which shows that $\psi_{i}^{1}$ is bijective, ending the proof of $(2)$.

$(3)$ For ${\underline{\lambda}}\in\Lambda_{r}^{i}(P)$ and ${\underline{\tau}}\in\Lambda_{s}^{j}(Q)$, for $1\leq j\leq i$, we have that the weak composition $\psi_{i}^{2}(P,Q)({\underline{\lambda}},{\underline{\tau}})=({\underline{\gamma}},{\underline{\delta}})$ satisfies the following conditions

1. (i)

${\underline{\gamma}}={\underline{\lambda}}$ belongs to $\Lambda_{r}^{i}(P)$, 2. (ii)

the weak composition ${\underline{\tau}}$ belongs to $\Lambda_{s}^{j}(Q)$, for some $1\leq j\leq i$. So, the sequence $\alpha_{Q}(d_{L(Q){-}\tau_{s}+1}^{p})+n\dots\alpha_{Q}(d_{L(Q)}^{p})+n$ is a word in the digits of $\{n+1,\dots,n+p\}$ such that each sequence appears at most $j$ times.

On the other hand, the sequence $\alpha_{P}(d_{L(P)-\gamma_{r}+1}^{n})\dots\alpha_{P}(d_{L(P)}^{n})$ is a word in $\{1,\dots,n\}$ such that some digit appears exactly $i$ times in it and no digit appears more than $i$ times.

The sequence $\omega_{n+p}^{P*_{\underline{\gamma}}Q}$, of level $n+p$ of $P*_{\underline{\gamma}}Q$, is equal to

[TABLE]

which shows that ${\underline{\delta}}=(\tau_{0},\dots,\tau_{s-1},\tau_{s}+\lambda_{r})$ belongs to $\Lambda_{s}^{i}(P*_{\gamma}Q)$. 3. (iii)

As ${\underline{\gamma}}={\underline{\lambda}}$ and ${\underline{\delta}}=(\tau_{0},\dots,\tau_{s-1},\tau_{s}+\lambda_{r})$, with $\tau_{s}>0$, we get that $\gamma_{r}<\delta_{s}$.

The map $({\underline{\gamma}},{\underline{\delta}})\mapsto({\underline{\gamma}},(\delta_{0},\dots,\delta_{s-1},\delta_{s}-\gamma_{r}))$ is the inverse map of $\psi_{i}^{2}(P,Q)$. $\square$

Theorem 4.3.9**.**

The binary operations $*_{0},\dots,*_{m}$ defined on ${\mathbb{K}}[\mbox{\it Dy}^{m}]$ satisfy the following relations

1. (1)

$x*_{i}(y*_{j}z)=(x*_{i}y)*_{j}z$, for $0\leq i<j\leq m$, 2. (2)

$x*_{i}(y*_{0}z+\dots+y*_{i}z)=(x*_{i}y+\dots+x*_{m}y)*_{i}z$, for $0\leq i\leq m$,

where $x,y,z$ are arbitrary elements of ${\mathbb{K}}[\mbox{\it Dy}^{m}]$.

• Proof. Clearly, it suffices to prove the relations for any Dyck paths $P$, $Q$ and $Z$. Suppose that $P\in\mbox{\it Dy}_{n}^{m}$, $Q=Q_{1}\times_{0}\dots\times_{0}Q_{r}\in\mbox{\it Dy}_{p}^{m}$ and $Z=Z_{1}\times_{0}\dots\times_{0}Z_{s}\in\mbox{\it Dy}_{q}^{m}$, where $Q_{1},\dots,Q_{r},Z_{1},\dots,Z_{s}$ are prime Dyck paths.

$(1)$ For $0\leq i<j\leq m$, applying a recursive argument on $s$ and Lemma 4.3.5 it is easy to see that,

[TABLE]

for any pair $({\underline{\lambda}},{\underline{\tau}})\in\Lambda_{r}^{i}(P)\times\Lambda_{s}^{j}(Q)$, where ${\underline{\delta}}=(\tau_{0},\dots,\tau_{s-1},\tau_{s}+\lambda_{r})$.

Applying the same notation than in Proposition 4.3.8, we get that

$P*_{{\underline{\lambda}}}(Q*_{\underline{\tau}}Z)=(P*_{{\underline{\lambda}}}Q)*_{\underline{\delta}}Z$ if, and only if, $\psi_{ij}(P,Q)({\underline{\lambda}},{\underline{\tau}})=({\underline{\lambda}},{\underline{\delta}})$.

The result follows applying point $(1)$ of Proposition 4.3.8.

$(2)$ We write ${\displaystyle\sum_{j=0}^{i}P*_{i}(Q*_{j}Z)=P*_{i}(Q*_{0}Z)+\sum_{j=1}^{i}P*_{i}(Q*_{j}Z)}$ and we work the terms on the right hand side separately.

$(a)$ Suppose that ${\underline{\tau}}\in\Lambda_{s}^{0}(Q)$, by Lemma 4.3.5 we get

[TABLE]

where $\underline{\tau}^{\prime}=(\tau_{0},\dots,\tau_{j_{\underline{\tau}}})$ and $Q_{r}*_{\underline{\tau}^{\prime}}(Z_{1}\times_{0}\dots\times_{0}Z_{j_{\underline{\tau}}})$ is prime.

Applying $P*_{{\underline{\lambda}}}$, we obtain that

[TABLE]

for the weak compositions ${{\underline{\lambda}}^{1}}=(\lambda_{0},\dots,\lambda_{r-1},\lambda_{r}+\dots+\lambda_{r+s{-}j_{\underline{\tau}}})$,

${\underline{\lambda}}^{2}=(\lambda_{r},\dots,\lambda_{r+s{-}j_{\underline{\tau}}})$, ${\underline{\tau}^{2}}=(\tau_{0},\dots,\tau_{j_{\underline{\tau}}-1},\tau_{j_{\underline{\tau}}}+\lambda_{r}+\dots+\lambda_{r+s{-}j_{\underline{\tau}}})$ and ${\underline{\delta}}=(\tau_{0},\dots,\tau_{j_{\underline{\tau}}-1},\tau_{j_{\underline{\tau}}}+\lambda_{r},\lambda_{r+1},\dots,\lambda_{r+s{-}j_{\underline{\tau}}})$.

The formula above implies that for any pair $({\underline{\lambda}},{\underline{\tau}})\in\Lambda_{r+s{-}j_{\underline{\tau}}}^{i}(P)\times\Lambda_{s}^{0}(Q)$, the elements $P*_{{\underline{\lambda}}}(Q*_{\underline{\tau}}Z)$ and $(P*_{\underline{\gamma}}Q)*_{\underline{\delta}}Z$ are equal whenever

[TABLE]

So, we have proved that

[TABLE]

where the sum is taken over all ${\underline{\gamma}}\in\displaystyle\bigcup_{j=i}^{m}\Lambda^{j}_{r}(P)$ and ${\underline{\delta}}\in\Lambda_{s}^{i}(P*_{\underline{\gamma}}Q)$, satisfying that $\delta_{s}\leq\gamma_{r}$.

$(b)$ Suppose that $({\underline{\lambda}},{\underline{\tau}})$ belongs to $\Lambda_{r}^{i}(P)\times({\displaystyle\bigcup_{j=1}^{i}\Lambda_{s}^{j}(Q)})$. We have that

[TABLE]

with $Q_{1},\dots,Q_{r-1},Q_{r}*_{\underline{\tau}}Z$ prime. So, computing

[TABLE]

where ${\underline{\lambda}}^{1}=(\lambda_{0},\dots,\lambda_{r-2},\lambda_{r-1}+\lambda_{r})$ and ${\underline{\delta}}=(\tau_{0},\dots,\tau_{s-1},\tau_{s}+\lambda_{r}).$

Using the notation of Proposition 4.3.8, we have proved the equality

[TABLE]

whenever $\psi_{i}^{2}(P,Q)({\underline{\lambda}},{\underline{\tau}})=({\underline{\gamma}},{\underline{\delta}})$. So, we get

[TABLE]

where the sum is taken over all $({\underline{\gamma}},{\underline{\delta}})\in\Lambda_{r}^{i}(P)\times\Lambda_{s}^{i}(P*_{\underline{\gamma}}Q)$ such that $\delta_{s}>\gamma_{r}$.

Finally, adding up $(a)$ and $(b)$, we get

[TABLE]

which ends the proof. $\square$

Theorem 4.3.9 asserts that the graded vector space ${\mathbb{K}}[\mbox{\it Dy}^{m}]$ spanned by the set of all $m$-Dyck paths, equipped with the operations $*_{i}$, is a $\mbox{\it Dyck}^{m}$ algebra, for all $m\geq 1$.

We now turn to prove that $({\mathbb{K}}[\mbox{\it Dy}^{m}];*_{0},\dots,*_{m})$ is in fact the free $\mbox{\it Dyck}^{m}$ algebra on one generator, that means ${\mathbb{K}}[\mbox{\it Dy}^{m}]$ is isomorphic to ${\mathfrak{D}}^{m}$.

Proposition 4.3.10**.**

Any element of $P\in\mbox{\it Dy}_{n}^{m}$ is of the form $R_{1}*_{i}R_{2}$, where $0\leq i\leq m$ and the sizes of $R_{1}$ and $R_{2}$ are strictly smaller than $n$, for $n\geq 1$.

• Proof. Suppose that $P=P_{1}\times_{0}P_{2}\times_{0}\dots\times_{0}P_{r}$, with $P_{i}$ prime, for $1\leq i\leq r$.

If $r>1$, then $P=P^{\prime}*_{0}P_{r}$, with $P^{\prime}:=P_{1}\times_{0}\dots\times_{0}P_{r-1}$, and the result is true.

If $P$ is prime, then $0\leq L_{1}(P)<m$. Let $X(P)=\{1<s_{1}<\dots<s_{k}=n\}$ be the set of integers satisfying that there exists at least one down steps in $\mathcal{DW}^{s_{j}}(P)$ of color $1$, and let $h_{j}>0$ be the number of down steps of color $1$ in $\mathcal{DW}^{s_{j}}(P)$, for $1\leq j\leq k$. We have that $h_{1}+\dots+h_{k}+L_{1}(P)=m$.

It is immediate to see that there exist $m$-Dyck paths $Q_{1},\dots,Q_{k}$, satisfying that

[TABLE]

where $l_{j}=h_{j}+\dots+h_{k}$, for $1\leq j\leq k$.

Let $R_{1}=(((\rho_{m}\times_{l_{1}}Q_{1})\times_{l_{2}}Q_{2})\dots)\times_{l_{k-1}}Q_{k-1}$. The size of $R_{1}$ is $n_{1}$, for some $0<n_{1}<n$, and the number of steps of color $1$ in $\mathcal{DW}^{n_{1}}(R_{1})$ is $h_{k-1}+h_{k}$. As $h_{k-1}\geq 1$, we have that $P=R_{1}*_{h_{k}}Q_{k}$, which ends the proof. $\square$

The following theorem states that the graded vector space $\mathcal{D}_{m}$ also describes the algebraic operad $\mbox{\it Dyck}^{m}$.

Theorem 4.3.11**.**

The free $\mbox{\it Dyck}^{m}$ algebra on one generator is isomorphic to $({\mathbb{K}}[\mbox{\it Dy}^{m}],*_{0},\dots,*_{m})$.

• Proof. As ${\mathbb{K}}[\mbox{\it Dy}^{m}]$ is a $\mbox{\it Dyck}^{m}$ algebra, there exists a unique homomorphism $\phi:{\mathfrak{D}}^{m}\longrightarrow{\mathbb{K}}[\mbox{\it Dy}^{m}]$ such that $\phi(|)$ is $\rho_{m}$, the unique $m$-Dyck path of size $1$. Proposition 4.3.10 implies that $\phi$ is surjective.

By Proposition 4.3.10, the subspace of homogeneous elements of degree $n$ of ${\mathbb{K}}[\mbox{\it Dy}^{m}]$ is generated by the subset $\mbox{\it Dyck}_{n}^{m}$ of $m$-Dyck paths of size $n$. Let ${\mathfrak{D}}_{n}^{m}$ be the subspace of elements of degree $n$ of ${\mathfrak{D}}^{m}$.

As $\phi$ is surjective, to prove that $\phi$ is an isomorphism it suffices to show that the dimension of the vector space ${\mathfrak{D}}_{n}^{m}$ is the number of elements of the set $\mbox{\it Dy}_{n}^{m}$, that is

[TABLE]

which was proved in Corollary 1.3.5. $\square$

4.4. Connection with the $m$-Tamari lattice

F. Bergeron extended the Tamari order to the sets $\mbox{\it Dy}_{n}^{m}$ of Dyck paths (see [7]) . Let us describe briefly the $m$-Tamari lattice $\mbox{\it Dy}_{n}^{m}$.

Let $P$ be an $m$-Dyck path. For any down step $d_{0}\in\mathcal{DW}(P)$ which is followed by an up step $u\in\mathcal{UP}(P)$, consider the excursion $P_{u}$ of $u$ in $P$ and its matching down step $w_{u}$ as described in Definition 4.1.4. Let $P_{(d_{0})}$ be the Dyck path obtained by removing $d_{0}$ and gluing the initial vertex of $u$ to the end of the step preceding $d_{0}$, and attaching $d_{0}$ at the final point of $w_{u}$. For example

It is immediate to see that $\alpha_{P_{d_{0}}}(d)=\alpha_{P}(d)$, for any $d\in\mathcal{DW}(P)$.

Definition 4.4.1**.**

The $m$-Tamari order on $\mbox{\it Dy}_{n}^{m}$ is the transitive relation spanned by the covering relation

[TABLE]

for any $d\in\mathcal{DW}(P)$ such that the final vertex of $d$ is the initial point of an up step $u\in\mathcal{UP}(P)$. We use the symbol $\lessdot$ for a covering relation.

The Hasse diagrams for $m=2$ and $n=1,2$ are

The goal of the present section is to show that the binary operations $*_{i}:{\mathbb{K}}[\mbox{\it Dy}_{n}^{m}]\otimes{\mathbb{K}}[\mbox{\it Dy}_{r}^{m}]\longrightarrow{\mathbb{K}}[\mbox{\it Dy}_{n+r}^{m}]$ are described in terms of the $m$-Tamari order.

Remark 4.4.2**.**

1. (1)

Let $Q$ be a prime Dyck path, for any pair of $m$-Dyck path $P$, we get

[TABLE]

in the $m$-Tamari lattice. 2. (2)

If $P<P^{\prime}$ in $\mbox{\it Dy}_{n_{1}}^{m}$ are such that $L(P)=L(P^{\prime})$, and $Q<Q^{\prime}$ in $\mbox{\it Dy}_{n_{2}}^{m}$, then

1. (a)

$P\times_{k}Q<P\times_{k}Q^{\prime}$, for any $0\leq k\leq L(P)$, 2. (b)

$P\times_{k}Q<P^{\prime}\times_{k}Q$, for any $0\leq k\leq L(P)$.

For the rest of the section, the $m$-Dyck path $Q$ is supposed to be a product $Q=Q_{0}\times_{0}\dots\times_{0}Q_{r}$, where all the $Q_{j}$’s are prime Dyck paths.

Lemma 4.4.3**.**

Let $P\in\mbox{\it Dy}_{n_{1}}^{m}$ and $Q\in\mbox{\it Dy}_{n_{2}}^{m}$ be two Dyck paths. Two weak compositions ${\underline{\lambda}}$ and ${\underline{\gamma}}$ in $\Lambda_{r}(P)$ satisfy that

[TABLE]

for $1\leq j\leq r$, if, and only if, $P*_{{\underline{\lambda}}}Q\leq P*_{\underline{\gamma}}Q$.

• Proof. If $Q$ is prime, the result follows from point $(1)$ of Remark 4.4.2. Suppose that $Q=Q_{1}\times_{0}\dots\times_{0}Q_{r}$, for $r>1$. A recursive argument shows that, for any pair of elements ${\underline{\lambda}}^{\prime}$ and ${\underline{\gamma}^{\prime}}$ in $\Lambda_{r-1}(P)$, we have that

[TABLE]

whenever $\lambda^{\prime}_{j}+\dots+\lambda^{\prime}_{r-1}\leq\gamma^{\prime}_{j}+\dots+\lambda^{\prime}_{r-1},$ for $1\leq j\leq r-1$.

So, we get

1. (a)

$P*_{{\underline{\lambda}}}Q=(P*_{{\underline{\lambda}}^{\prime}}(Q_{1}\times_{0}\dots\times_{0}Q_{r-1}))\times_{\lambda_{r}}Q_{r}$, 2. (b)

$P*_{\underline{\gamma}}Q=(P*_{\underline{\gamma}^{\prime}}(Q_{1}\times_{0}\dots\times_{0}Q_{r-1}))\times_{\gamma_{r}}Q_{r}$,

where ${\underline{\lambda}}^{\prime}=(\lambda_{0},\dots,\lambda_{r-1},\lambda_{r-1}+\lambda_{r})$ and $\underline{\gamma}^{\prime}=(\gamma_{0},\dots,\gamma_{r-1},\gamma_{r-1}+\gamma_{r})$. The recursive hypothesis implies that

[TABLE]

and, using that $\lambda_{r}\leq\gamma_{r}$, we obtain $P*_{{\underline{\lambda}}}Q\leq P*_{\underline{\gamma}}Q$.

Conversely, suppose that $P*_{{\underline{\lambda}}}Q\leq P*_{\underline{\gamma}}Q$. Point $(3)$ of Remark 4.4.2 implies that

[TABLE]

for $1\leq j\leq r$, which ends the proof. $\square$

Notation 4.4.4**.**

For any $m$-Dyck path $P$ of size $n$ and any $0\leq i\leq m$, let

1. (1)

$c_{i}(P)$ be the minimal number of elements such that the word

[TABLE]

contains $i$ times an integer in $\{1,\dots,n\}$ and no integer more than $i$ times, 2. (2)

$C_{i}(P)$ be the maximal integer such that the word

[TABLE]

contains at least one integer repeated $i$ times and no integer repeated $i+1$ times.

Let $P\in\mbox{\it Dy}_{n_{1}}^{m}$ and $Q\in\mbox{\it Dy}_{n_{2}}^{m}$ be two Dyck paths. For any integer $0\leq i\leq m$, let $P/_{i}Q$ and $P\backslash_{i}Q$ be the Dyck paths defined as follows

1. (a)

$P/_{i}Q:=P\times_{c_{i}(P)}Q$, 2. (b)

$P\backslash_{i}Q:=(P\times_{L(P)}(Q_{1}\times_{0}\dots\times_{0}Q_{r-1}))\times_{C_{i}(P)}Q_{r}$.

Proposition 4.4.5**.**

For any pair of Dyck paths $P\in\mbox{\it Dy}_{n_{1}}^{m}$ and $Q\in\mbox{\it Dy}_{n_{2}}^{m}$ and any integer $0\leq i\leq m$, the product $*_{i}$ is given in terms of the $m$-Tamari order by the following formula

[TABLE]

• Proof. Suppose that $Q=Q_{1}\times_{0}\dots\times_{0}Q_{r}$, with all the $Q_{i}$’s prime and that ${\underline{\lambda}}\in\Lambda_{r}^{i}(P)$.

The weak composition ${\underline{\lambda}}=(\lambda_{0},\dots,\lambda_{r})$ satisfies that $c_{i}(P)\leq\lambda_{r}\leq C_{i}(P)$ and ${\displaystyle\sum_{j=0}^{r}\lambda_{i}=L(P)}$. As we have that

1. (a)

$P/_{i}Q=P*_{(L(P){-}c_{i}(P),0,\dots,0,c_{i}(P))}Q$, and 2. (b)

$P\backslash_{i}Q=P*_{(0,\dots,0,L(P){-}C_{i}(P),C_{i}(P))}Q$,

applying Lemma 4.4.3, it is easily seen that $P/_{i}Q\leq P*_{{\underline{\lambda}}}Q\leq P\backslash_{i}Q$.

Recall that, whenever $R<S$ in the Tamari lattice, the set $\mathcal{DW}(R)$ of down steps of $R$ is identified with the set $\mathcal{DW}(S)$. For any $d\in\mathcal{DW}(P)$ the levels of $d$ in $R$ and in $S$ are different but $\alpha_{R}(d)=\alpha_{S}(d)$.

Note that the unique down steps which have different levels in the Dyck paths $P/_{i}Q$ and $P\backslash_{i}Q$ are colored by the set of integers $\{1,\dots,n_{1}\}$. So, for any $P/_{i}Q\leq Z\leq P\backslash_{i}Q$ and any $1\leq l\leq r$, we get that

[TABLE]

Define

[TABLE]

The arguments above show that

1. (1)

$c_{i}\leq\lambda_{r}\leq C_{i}$, 2. (2)

$c_{i}\leq\lambda_{j}+\dots+\lambda_{r}\leq L(P)$, for $1\leq j\leq r-1$, 3. (3)

$0\leq L_{n_{1}}(Z)\leq L(P)-c_{i}.$

From $(4.4.4)$, we get that $Z=P*_{{\underline{\lambda}}}Q$.

Lemma 4.4.3 and $P/_{i}Q\leq P*_{{\underline{\lambda}}}Q\leq P\backslash_{i}Q$ imply that ${\underline{\lambda}}\in\Lambda_{r}^{i}(P)$. $\square$

Let us define the product $*$ on ${\mathbb{K}}[\mbox{\it Dy}^{m}]$ as the sum $*:=*_{0}+\dots+*_{m}$. It is not difficult to see, using Proposition 4.4.5, that

[TABLE]

Example 4.4.6**.**

Consider the Dyck paths $P=(1,3)$ and $Q=(2,2)$ in $\mbox{\it Dy}_{2}^{2}$, the following diagram describes the Tamari interval $I_{P*Q}$ of all $Z\in\mbox{\it Dy}_{4}^{2}$ such that $P*Q={\displaystyle\sum_{Z\in I_{P*Q}}Z}$.

The Dyck paths in red are the terms of $P*_{0}Q$, the ones in green are the terms of $P*_{1}Q$, and the ones in blue are the terms appearing in $P*_{2}Q$.