Weighted $\mathsf{P}-$partitions enumerator

TL;DR

This paper introduces a weighted integer points enumerator for extended permutohedra, connecting it to quasisymmetric functions and refining existing enumerators for poset cones, with implications for combinatorial enumeration.

Contribution

It defines a new weighted enumerator linked to extended permutohedra and shows it as a universal morphism from the Hopf algebra of posets, refining Gessel's P-partitions enumerator.

Findings

The enumerator generalizes Gessel's P-partitions enumerator.

It is a quasisymmetric function derived from a universal morphism.

The principal specialization yields the f-polynomial of the extended permutohedron.

Abstract

To an extended permutohedron we associate the weighted integer points enumerator, whose principal specialization is the $f$-polynomial. In the case of poset cones it refines Gessel's $\mathsf{P}$-partitions enumerator. We show that this enumerator is a quasisymmetric function obtained by universal morphism from the Hopf algebra of posets.

Equations

$$M_{\alpha }=\sum _{i_1<\cdots <i_k}{x}_{i_1}^{{\alpha }_1}\cdots x_{i_k}^{{\alpha }_k}.$$

$${\zeta }_{\mathcal{Q}}(M_{\alpha })=\{\begin{array}{ll}1,& \text{if }\alpha =(n)\text{ for }n\in \mathbb{N},\\ 0,& \text{otherwise.}\end{array}$$

$$\Psi \circ {\zeta }_{\mathcal{Q}}=\zeta .$$

$${\zeta }_{\alpha }(h)={\zeta }^{\otimes k}\circ (p_{{\alpha }_1}\otimes \cdots \otimes p_{{\alpha }_k})\circ {\Delta }^{k-1}(h),$$

$${ps}^1(F)(m)=F(\underset{m}{\underbrace{1,\dots ,1}},0,0\dots ).$$

$${ps}^1(M_{\alpha })(m)=\left(\genfrac{}{}{0ex}{}{m}{k(\alpha )}\right),$$

$${ps}^1(M_{\alpha })(-1)=\left(\genfrac{}{}{0ex}{}{-1}{k(\alpha )}\right)=(-1{)}^{k(\alpha )}.$$

$$L_{\alpha }=\sum _{\alpha ⪯\beta }{M}_{\beta }.$$

$$i{<}_{\mathsf{P}}j\text{implies}f(i)<f(j).$$

$$F_{\mathsf{P}}(x)=\sum _{f\in \mathcal{A}(\mathsf{P})}{x}_{f(1)}{x}_{f(2)}\cdots x_{f(n)}.$$

$$F_{\mathsf{P}}(x)=L_{\alpha (\mathsf{P})},$$

$$F_{\mathsf{P}}(x)=\sum _{\mathsf{l}\in \mathcal{L}(\mathsf{P})}{F}_{\mathsf{l}}(x),$$

$$P{e}^{n-1}=\mathrm{conv}\{(a_{\sigma (1)},a_{\sigma (2)},\dots ,a_{\sigma (n)}):\sigma \in {\mathfrak{S}}_n\},$$

$$\sum _{i\in C_1\cup C_2\cup \cdots \cup C_k}{x}_i=a_1+a_2+\cdots +a_{|C_1\cup C_2\cup \cdots \cup C_k|},$$

$$\mathcal{F}:\varnothing =F_0\subset F_1\subset \cdots \subset F_{n-d-1}\subset F_{n-d}=[n],$$

$$\dim (\mathcal{F})=n-|\mathcal{F}|.$$

Full text

Keywords: generalized permutohedron, quasisymmetric function, poset, combinatorial Hopf algebra

1 Introduction

In the seminal paper of Aguiar, Bergeron and Sottile [ABS] the notion of combinatorial Hopf algebra was introduced. They explained the ubiquity of quasisymmetric functions as generating functions in enumerative combinatorics. More recently a geometric meaning of quasisymmetric enumerators is attributed. It is based on a class of convex polytopes called generalized permutohedra. The integer points enumerator associated to a generalized permutohedron is a quasisymmetric function. It was defined, and studied in the case of matroid base polytopes, by Billera, Jia and Reiner in [BJR], and in the case of nestohedra by Grujić in [G]. More subtle generalization which takes into account the face structure of a generalized permutohedron is introduced and studied in [GPS]. In this paper we consider the extended generalized permutohedra and the special case of poset cones. We prove that the integer points enumerator associated to a poset cone coincides with the universal morphism from the Hopf algebra of posets to quasisymmetric functions. The specialization for $q=0$ is the Gessel enumerator of ${\mathsf{P}}$-partitions which attributes the geometric meaning to this classical function.

In sections 2 we review necessary facts about quasisymmetric functions and ${\mathsf{P}}$-partitions enumerators. In section 3 we introduce the integer points enumerator $F_{q}(P)$ for an extended generalized permutohedron $P$, which is a weighted quasisymmetric function, in the same manner as in the case of generalized permutohedra provided in [GPS]. The parameter $q$ reflects the rank function of the face lattice $L(P)$. To a poset ${\mathsf{P}}$ is associated the poset cone $C(\mathsf{P})$, which is an extended generalized permutohedron and our construction produces a weighted quasisymmetric function $F_{q}(C(\mathsf{P}))$. In section 4 we prove Theorem LABEL:jakobitno, the first main result of the paper, which states that the weighted quasisymmetric function $F_{q}(C(\mathsf{P}))$, constructed geometrically, has an algebraic meaning as the universal morphism from a certain combinatorial Hopf algebra of posets $\mathcal{P}$ to the Hopf algebra $QSym$ of quasisymmetric functions. This result is analogous to the previous results for simple graphs [G1], matroids and building sets [GPS], and spreads their validity to the case of extended generalized permutohedra. The main theorem is followed by various examples, and statements about behavior of the enumerator of the opposite poset and under the action of the antipode. In Theorem LABEL:druga in section 5, it is shown that for a well labelled poset ${\mathsf{P}}$ and $q=0$ our enumerator specializes to the classical Gessel’s ${\mathsf{P}}$-partitions enumerator. We also provide an example of posets with the same ${\mathsf{P}}$-partitions enumerators but which are distinguished by corresponding weighted quasisymmetric enumerators.

2 Quasisymmetric functions

A composition $\alpha$ of a positive integer $n$, $\alpha\models n$, is an ordered list $(\alpha_{1},\ldots,\alpha_{k})$ of positive integers such that $\alpha_{1}+\cdots+\alpha_{k}=n$. The monomial quasisymmetric function $M_{\alpha}$ indexed by the composition $\alpha$ is an element of the commutative algebra of formal power series in the countable ordered set of variables $\mathbb{x}=(x_{1}<x_{2}<x_{3}<\cdots)$ defined by

[TABLE]

The algebra of quasisymmetric functions $\mathcal{Q}Sym$, spanned by $M_{\alpha}$ when $\alpha$ runs over all compositions, is a subalgebra of the algebra of formal power series. The algebra $\mathcal{Q}Sym$ is a graded, connected Hopf algebra (see [GR], Proposition 5.8). The homogeneous component $QSym_{n}$ is spanned by $\{M_{\alpha}\}_{\alpha\models n}$. Let $\zeta_{\mathcal{Q}}:\mathcal{Q}Sym\rightarrow\mathbb{k}$ be a linear multiplicative functional defined on the monomial basis by

[TABLE]

The Hopf algebra $\mathcal{Q}Sym$ equipped with the character $\zeta_{Q}$ is the terminal object in the category of combinatorial Hopf algebras.

Theorem 2.1** ([ABS], Theorem 4.1).**

For a combinatorial Hopf algebra $(\mathcal{H},\zeta)$ there is a unique morphism of graded Hopf algebras $\Psi:H\rightarrow\mathcal{Q}Sym$ such that

[TABLE]

For a homogeneous element $h$ of degree $n$ the coefficients $\zeta_{\alpha}(h)$, $\alpha=(\alpha_{1},\ldots,\alpha_{k})\models n$ of $\Psi(h)$ in the monomial basis are given by

[TABLE]

*where $p_{i}$ is the projection of $\mathcal{H}$ on the $i$-th homogeneous component $\mathcal{H}_{i}$ and $\Delta^{k-1}$ is the *$(k-1)-$fold coproduct map of $\mathcal{H}$.

For $F\in\mathcal{Q}Sym$ and $m\in\mathbb{N}$, let $\mathbb{ps}^{1}$ denotes the principal specialization

[TABLE]

We have

[TABLE]

where $k(\alpha)$ is the number of parts of $\alpha=(\alpha_{1},\ldots,\alpha_{k}).$ Specially, for $m=-1$ we have

[TABLE]

For a composition $\alpha=(\alpha_{1},\ldots,\alpha_{k})\models n$ let $D(\alpha)\subseteq[n-1]$ be a subset defined by $D(\alpha)=\{\alpha_{1},\alpha_{1}+\alpha_{2},\ldots,\alpha_{1}+\alpha_{2}+\cdots+\alpha_{k-1}\}$. We say that $\alpha\models n$ refines $\beta\models n$, and write $\beta\preceq\alpha$, if $D(\beta)\subseteq D(\alpha).$

Another important basis of $\mathcal{Q}Sym$ is the basis of fundamental quasisymmetric functions defined by

[TABLE]

2.1 Quasisymmetric enumerator of $\mathsf{P}-$partitions

A labelled poset $\mathsf{P}$ is a poset on some finite subset of positive integers. A $\mathsf{P}-$partition is a function $f:\mathsf{P}\rightarrow\mathbb{N}$ such that

1. $\bullet$

$i<_{\mathsf{P}}j$ and $i<_{\mathbb{Z}}j$ implies $f(i)\leq f(j)$, 2. $\bullet$

$i<_{\mathsf{P}}j$ and $i>_{\mathbb{Z}}j$ implies $f(i)<f(j).$

Definition 2.2**.**

A poset $\mathsf{P}$ is a well labelled poset if $i<_{\mathsf{P}}j$ implies $i>_{\mathbb{Z}}j.$ In that case $\mathsf{P}-$partition is a function $f:\mathsf{P}\rightarrow\mathbb{N}$ such that

[TABLE]

Denote by $\mathcal{A}(\mathsf{P})$ the set of all $\mathsf{P}-$partitions. Define the enumerator of $\mathsf{P}-$partitions by

[TABLE]

Proposition 2.3** ([GR], Proposition 5.18).**

For a totally ordered labelled poset $\mathsf{P}=\{i_{1}<_{\mathsf{P}}i_{2}<_{\mathsf{P}}\cdots<_{\mathsf{P}}i_{n}\}$ the enumerator of $\mathsf{P}$-partitions is equal to the fundamental quasisymmetric function

[TABLE]

where $\alpha(\mathsf{P})\models n$ is a composition such that $D(\alpha(\mathsf{P}))=\{j:i_{j}>_{\mathbb{Z}}i_{j+1}\}.$

Theorem 2.4** ([GR], Theorem 5.19).**

For a labelled poset ${\mathsf{P}}$,

[TABLE]

where the sum is over the set $\mathcal{L}({\mathsf{P}})$ of all linear extensions $\mathsf{l}$ of ${\mathsf{P}}$.

Example 2.5**.**

Figure 1 presents two posets with their enumerators of ${\mathsf{P}}$-partitions. Note that the poset ${\mathsf{P}}_{1}$ is not well labelled, while the poset ${\mathsf{P}}_{2}$ is.

3 Extended generalized permutohedra

A standard $(n-1)-$dimensional permutohedron $Pe^{n-1}$ is the convex hull of the orbit of a point $(a_{1},a_{2},\ldots,a_{n})\in\mathbb{R}^{n}$ with increasing coordinates $a_{1}<a_{2}<\cdots<a_{n}$

[TABLE]

where $\mathfrak{S}_{n}$ is the permutation group of $[n]$.

Proposition 3.1** ([P], Proposition 2.6).**

The $d-$dimensional faces of $Pe^{n-1}$ are in one-to-one correspondence with set compositions $\mathcal{C}=C_{1}|C_{2}|\cdots|C_{n-d}$ of $[n].$ The face corresponding to the set composition $\mathcal{C}$ is given by the $n-d$ linear equations

[TABLE]

for $1\leq k\leq n-d$.

A set composition $\mathcal{C}=C_{1}|C_{2}|\cdots|C_{n-d}$ defines the flag $\mathcal{F}$ of subsets

[TABLE]

of the length $|\mathcal{F}|=n-d$, where $F_{i}=C_{1}\cup C_{2}\cup\cdots\cup C_{i}$, for $1\leq i\leq n-d$. There is an obvious order reversing one$-$to$-$one correspondence between the face lattice of the permutohedron $L(Pe^{n-1})$ and the lattice of flags of subsets of the set $[n].$ Using this correspondence we will label faces of the standard permutohedron $Pe^{n-1}$ by flags of subsets of $[n]$. We have

[TABLE]

The normal fan $\mathcal{N}(Pe^{n-1})$ of the standard permutohedron $Pe^{n-1}$ is the fan of the braid arrangement, given by hyperplanes $\{x_{i}=x_{j}\}_{1\leq i<j\leq n}$, in $\mathbb{R}^{n}$. The cones of the braid arrangement fan are called braid cones. The braid cone $C_{\mathcal{F}}$ at the face $\mathcal{F}$ is determined by

1. $\bullet$

$x_{p}=x_{q}$ if $p,q\in F_{i+1}\setminus F_{i}$, for some $0\leq i\leq|\mathcal{F}|-1,$ 2. $\bullet$

$x_{p}\leq x_{q}$ if $p\in F_{i}\setminus F_{i-1}$ and $q\in F_{i+1}\setminus F_{i}$, for some $1\leq i\leq|\mathcal{F}|-1$.

We have $\dim(C_{\mathcal{F}})=|\mathcal{F}|$. The fan $\mathcal{N}_{1}$ is a refinement of $\mathcal{N}_{2}$ (or $\mathcal{N}_{2}$ is a coarsement of $\mathcal{N}_{1}$) if every cone of $\mathcal{N}_{1}$ is contained in a cone in $\mathcal{N}_{2}$ (or if every cone in $\mathcal{N}_{2}$ is a union of cones of $\mathcal{N}_{1}$).