Noncommutative unicellular LLT polynomials

TL;DR

This paper extends the relationship between unicellular LLT polynomials and graph chromatic polynomials to a noncommutative setting, broadening understanding of their algebraic structure.

Contribution

It introduces a noncommutative version of the $(t-1)$-transform relationship for unicellular LLT polynomials within combinatorial Hopf algebras.

Findings

Established a noncommutative analogue of the $(t-1)$-transform property.

Connected unicellular LLT polynomials to noncommutative symmetric functions.

Extended combinatorial Hopf algebra frameworks to include these polynomials.

Abstract

It is known that unicellular LLT polynomials are related to the quasi-symmetric chromatic polynomials of certain graphs by the $(t-1)$-transform of symmetric functions. We investigate the extension of this transformation to various combinatorial Hopf algebras and prove a noncommutative version of this property.

Equations

$${\mathrm{X}}_G(t,X)=(t-1{)}^{-n}{\mathrm{LLT}}_G(t,(t-1)X).$$

$${\mathrm{X}}_G(t,X)=\sum _{c\in C(G)}{t}^{{\mathrm{asc}}_G(c)}{x}_{c_1}{x}_{c_2}\cdots x_{c_n}=\sum _{c\in \mathrm{PC}(\mathrm{G})}{t}^{{\mathrm{asc}}_G(c)}{M}_{\mathrm{ev}(c)}(X),$$

$${\mathrm{LLT}}_G(t,X)=\sum _{u\in P{W}_n}{t}^{{\mathrm{asc}}_G(u)}{M}_{\mathrm{ev}(u)}(X),$$

$${\mathbf{M}}_u:=\sum _{\mathrm{pack}(w)=u}w.$$

$$\begin{array}{rl}{\mathbf{M}}_{13132}=& 13132+14142+14143+24243\\ & +15152+15153+25253+15154+25254+35354.\end{array}$$

$$M_I:=\sum _{j_1<j_2\dots <j_r}{x}_{j_1}^{i_1}{x}_{j_2}^{i_2}\cdots x_{j_r}^{i_r},$$

$${\mathbf{G}}_{\sigma }(A)=\sum _{\mathrm{std}(w)=\sigma }w,$$

$${\mathbf{G}}_{\sigma }=\sum _{\mathrm{std}(u)=\sigma }{\mathbf{M}}_u,$$

$${\mathbf{N}}_u\mapsto {\mathbf{F}}_{\mathrm{std}(u)}.$$

$${\mathbf{M}}_u(AB)=\sum _{\mathrm{pack}\left(\genfrac{}{}{0ex}{}{v}{w}\right)=u}{\mathbf{M}}_v(A){\mathbf{M}}_w(B),$$

$${\mathbf{N}}_u\ast {\mathbf{N}}_v={\mathbf{N}}_{\mathrm{pack}\left(\genfrac{}{}{0ex}{}{u}{v}\right)},$$

$$S^I\mapsto {\hat{S}}^I=\sum _{\mathrm{ev}(u)=I}{\mathbf{N}}_u.$$

$${\hat{S}}^I\mapsto {\iota }^{\ast }({\mathbf{N}}_{1^{i_1}}\cdots {\mathbf{N}}_{1^{i_r}})={\mathbf{F}}_{12\dots i_1}\cdots {\mathbf{F}}_{12\dots i_r}=S^I=\sum _{\mathrm{Des}(\sigma )\subseteq \mathrm{Des}(I)}{\mathbf{G}}_{\sigma },$$

$${\mathbf{N}}_u\cdot \sigma :={\mathbf{N}}_{u\sigma },\text{where}u\sigma =u_{\sigma (1)}{u}_{\sigma (2)}\cdots u_{\sigma (n)}.$$

$${\mathbf{N}}_{u\sigma }\ast {\hat{S}}^I=({\mathbf{N}}_u\ast {\hat{S}}^I)\cdot \sigma .$$

$$\mathrm{pack}\left(\genfrac{}{}{0ex}{}{u\tau }{v}\right)=\mathrm{pack}\left(\genfrac{}{}{0ex}{}{u}{v{\tau }^{-1}}\right)\cdot \tau .$$

$${\iota }^{\ast }({\hat{S}}^I\ast {\mathbf{N}}_v)$$

$$=\sum _{\mathrm{ev}(u)=I}{\mathbf{F}}_{\mathrm{std}\left(\genfrac{}{}{0ex}{}{u}{v}\right)}$$

$$=\sum _{\mathrm{ev}(u)=I}{\mathbf{F}}_{\mathrm{std}\left(\genfrac{}{}{0ex}{}{u{\sigma }^{-1}}{v{\sigma }^{-1}}\right)\circ \sigma }$$

$$=\sum _{\mathrm{ev}(u)=I}{\mathbf{F}}_{\mathrm{std}(u{\sigma }^{-1})\circ \sigma }$$

$$=S^I\ast {\mathbf{F}}_{\sigma }$$

$$={\iota }^{\ast }({\hat{S}}^I)\ast {\iota }^{\ast }({\mathbf{N}}_v).$$

$$S^I\ast S^J={\iota }^{\ast }({\hat{S}}^I\ast {\hat{S}}^J)=\sum _{M\in \mathrm{Mat}(I,J)}{S}^M,$$

$${\sigma }_1(XA)=\overrightarrow{\prod _{x\in X}}\overrightarrow{\prod _{a\in A}}(1-xa{)}^{-1}=\sum _I{M}_I(X)S^I(A)=\sum _I{U}_I(X)V_I(A)$$

$${\sigma }_1(XTA)=\sum _I{M}_I(XT)S^I(A)=\sum _I{M}_I(X)S^I(TA).$$

$$XT=\frac{X|}{|1-t}\text{and}TA=\frac{|A}{1-t|}.$$

$$M_I(X)\mapsto M_I\left(\frac{X|}{|1-t}\right)\text{and}{S}^I(A)\mapsto S^I\left(\frac{|A}{1-t|}\right)$$

$$M_I(X)\mapsto M_I(X(1-t))\text{and}{S}^I(A)\mapsto S^I\left((1-t)A\right)$$

$$M_I\mapsto {\chi }_T(M_I)=:M_I(T)$$

$$S_n(TA):={\chi }_T(S_n(XA))=\sum _{I\models n}{M}_I(T)S^I(A)$$

$$M_I(XT)=\sum _J⟨M_I(X),S^J(TA)⟩M_J(X),$$

$$⟨M_I(X),S^J(TA)⟩$$

$$=\{\begin{array}{ll}{M}_{I_1}(T)\cdots M_{I_s}(T)& \text{if I=I1⋯Is with Ik⊨jk}\\ 0& \text{otherwise.}\end{array}$$

Full text

1. Introduction

In their proof of the shuffle conjecture [3], Carlsson and Mellit obtain a remarkable relation between unicellular LLT polynomials and the quasi-symmetric chromatic polynomials [24] of certain graphs, namely

[TABLE]

The graphs $G$ are simple undirected graphs with vertices labelled $1,\ldots,n$, characterized by the property that if there is an edge $\{i,j\}$ with $i<j$, then all the $\{i^{\prime},j^{\prime}\}$ with $i\leq i^{\prime}<j^{\prime}\leq j$ are also edges of $G$. The number of such graphs is the Catalan number $c_{n}$. These are the incomparability graphs of certain posets $P$, known as unit interval orders [24].

Let $V(G)$ and $E(G)$ denote respectively the sets of vertices and edges of $G$. A coloring of $G$ is a map $c:\ V(G)\rightarrow{\mathbb{N}}^{*}$, which can be identified with a word $c_{1}c_{2}\cdots c_{n}$. A coloring is proper if $c_{i}\not=c_{j}$ whenever $\{i,j\}\in E(G)$. We denote by $C(G)$ the set of proper colorings of $G$. The chromatic quasi-symmetric function of $G$ expands in the $M$ basis of ${\it QSym}$ as [24]

[TABLE]

where $\rm PC\,(G)$ denotes the set of proper packed colorings, ${\rm asc}_{G}(c)$ is the number of edges $\{i<j\}$ such that $c_{i}<c_{j}$, and ${\rm ev}(c)$ is the evaluation (or content) of $c$, that is, the composition recording the number of occurences of each value of $c$. It can be shown that for the above graphs, ${\rm X}_{G}(t)$ is actually a symmetric function [24].

On another hand, LLT polynomials are $t$-analogues of products of skew Schur functions [19, 14]. By interpreting $s_{1}$ as $s_{\lambda/\mu}$ in various ways, one may obtain different $t$-analogues of the characteristic $s_{1}^{n}$ of the regular representation of ${\mathfrak{S}}_{n}$. These can be parametrized by the same graphs as above, and their expression given by the dinv statistic of Haglund, Haiman and Loehr [14] can be rephrased as

[TABLE]

where $u$ runs now over all packed words of length $n$, regarded as colorings of $G$.

Therefore, Equation (1) tells us that the transformation $(t-1)X$ just eliminates the improper colorings, a fact which is far from obvious. A “pedestrian” proof can be found in the appendix of [13]. The aim of this note is to provide a conceptual explanation (and a generalization) in terms of combinatorial Hopf algebras.

2. Word quasi-symmetric functions

Let $A=\{a_{1}<a_{2}<\dots\}$ be a totally ordered alphabet. The packed word $u=\operatorname{pack}(w)$ associated with a word $w\in A^{*}$ is a word over the alphabet of positive integers, obtained by the following process. If $b_{1}<b_{2}<\ldots<b_{r}$ are the letters occuring in $w$, $u$ is the image of $w$ by the homomorphism $b_{i}\mapsto i$.

A word of positive integers $u$ is said to be packed if $\operatorname{pack}(u)=u$. We denote by ${\rm PW}$ the set of packed words. With such a word, we associate the “polynomial”

[TABLE]

For example, with $A=\{1<2<3<4<5\}$,

[TABLE]

The evaluation ${\rm ev}(w)$ of a word $w$ is the sequence whose $i$-th term $|w|_{a_{i}}$ is the number of times the letter $a_{i}$ occurs in $w$, regarded as a finite integer vector by removing the trailing zeros.

Let ${\mathbb{K}}$ be a field of characteristic [math], assumed to contain all formal series in the formal parameter $t$ used in the sequel.

Under the abelianization $\chi:\ {\mathbb{K}}\langle A\rangle\rightarrow{\mathbb{K}}[X]$, the ${\bf M}_{u}$ are mapped to the monomial quasi-symmetric functions

[TABLE]

where $I=(|u|_{a})_{a\in A}=(i_{1},\ldots,i_{r})$ is the evaluation vector of $u$.

The ${\bf M}_{u}$ span a subalgebra of ${\mathbb{K}}\langle A\rangle$, called ${\bf WQSym}$ for Word Quasi-Symmetric functions, consisting of the invariants of the noncommutative version of Hivert’s quasi-symmetrizing action [5], which is defined by $\sigma\cdot w=w^{\prime}$ where $w^{\prime}$ is such that ${\rm std}(w^{\prime})={\rm std}(w)$ and $\chi(w^{\prime})=\sigma\cdot\chi(w)$, where ${\rm std}$ stands for the usual standardization algorithm, namely the algorithm that sends any word to the permutation having the same inversions. Hence, two words are in the same ${\mathfrak{S}}(A)$-orbit iff they have the same packed word.

When $A$ is infinite, ${\mathbb{K}}\langle A\rangle$ is interpreted as the algebra of formal series of bounded degree. Exactly as in the case of symmetric or quasi-symmetric functions, ${\bf WQSym}$ acquires then the structure of a Hopf algebra, with the natural coproduct given by the ordinal sum of mutually commuting alphabets.

The coproduct $A+B$ is indeed well-defined on ${\bf WQSym}$ and allows to consider its graded dual ${\bf WQSym}^{*}$. We shall denote by ${\bf N}_{u}\in{\bf WQSym}^{*}$ the dual basis of ${\bf M}_{u}$.

The algebra ${\bf FQSym}$ (Free Quasi-symmetric functions) may be defined as the subalgebra of ${\mathbb{K}}\langle A\rangle$ spanned by the

[TABLE]

where $\sigma$ runs over all permutations. It is also a Hopf algebra for the same coproduct. It is self-dual, and the dual basis ${\bf F}_{\sigma}={\bf G}_{\sigma}^{*}$ can be identified with ${\bf G}_{\sigma^{-1}}$. It is isomorphic to the Malvenuto-Reutenauer Hopf algebra [21, 4].

There is therefore an inclusion of Hopf algebras $\iota:\ {\bf FQSym}\hookrightarrow{\bf WQSym}$ given by

[TABLE]

whose dual is the projection $\iota^{*}:\ {\bf WQSym}^{*}\twoheadrightarrow{\bf FQSym}$

[TABLE]

Let $AB$ be the alphabet $\{ab|a\in A,\ b\in B\}$ endowed with the lexicographic order on the pairs $(a,b)$. It is easy to check that the coproduct $\delta:\ f\mapsto f(AB)$ is also well-defined: packing with respect to the lexicographic order makes sense, and

[TABLE]

where $\binom{u}{v}$ denotes the word in biletters $\binom{u_{i}}{v_{i}}$, lexicographically ordered with priority to the top letter.

The dual of the coproduct $AB$ is an internal product on each homogeneous component ${\bf WQSym}^{*}_{n}$ given by

[TABLE]

In this way, ${\bf WQSym}_{n}^{*}$ gets identified with the Solomon-Tits algebra of ${\mathfrak{S}}_{n}$ [22] for all $n$.

There is also a Hopf embedding of ${\bf Sym}$ (Noncommutative symmetric functions) into ${\bf WQSym}^{*}$ given by $S_{n}\mapsto\hat{S}_{n}:={\bf N}_{1^{n}}$:

[TABLE]

For example, $\hat{S}^{21}={\bf N}_{112}+{\bf N}_{121}+{\bf N}_{211}$.

Under the projection $\iota^{*}$,

[TABLE]

where $\operatorname{Des}(\sigma)$ denotes the descent set of $\sigma$, and $\operatorname{Des}(I)$ the set encoded by the composition $I$.

This projection is compatible with the internal products: on ${\bf Sym}$, the internal product is defined as dual to the coproduct $XY$ of $QSym$ [11, 21]. By definition, it maps $f(AB)$ to $f(XY)$. On ${\bf FQSym}$, the internal product on the ${\bf F}$-basis is ordinary composition: ${\bf F}_{\sigma}*{\bf F}_{\tau}={\bf F}_{\sigma\circ\tau}$, so that on the ${\bf G}$-basis, ${\bf G}_{\sigma}*{\bf G}_{\tau}={\bf G}_{\tau\circ\sigma}$.

Now, although the $*$ product of ${\bf WQSym}^{*}$ does not coincide with composition on permutations, we have the following compatibility.

Lemma 2.1**.**

Define a right action of ${\mathfrak{S}}_{n}$ on ${\bf WQSym}_{n}^{*}$ by

[TABLE]

Then, for any $I\vDash n$

[TABLE]

*Proof – *If $\hat{S}^{I}$ contains ${\bf N}_{v}$, it contains ${\bf N}_{v\tau}$ for all permutations $\tau$, and

[TABLE]

For example, with $u=111122$, $v=212211$, $\tau=451623$, we have $u\tau=121211$, $v\tau^{-1}=211212$, $\operatorname{pack}{\binom{121211}{212211}}=232411$, $\operatorname{pack}{\binom{111122}{211212}}=211234$, and $211234\tau=232411$.

This implies that $(f\cdot\sigma)*g=(f*g)\cdot\sigma$ for all $f\in{\bf WQSym}^{*}_{n},g\in{\bf Sym}_{n}$ and $\sigma\in{\mathfrak{S}}_{n}$.

Remark 2.2**.**

A similar argument actually proves the existence of the descent algebra. If $\sigma={\rm std}(v)$,

[TABLE]

This implies in particular that in ${\bf FQSym}$,

[TABLE]

where ${\rm Mat}(I,J)$ denotes the set of nonnegative integer matrices with row-sums vector $I$ and column-sums vector $J$ (cf. [10]). Hence, the $S^{I}=\sum_{\operatorname{Des}(\sigma^{-1})\subseteq\operatorname{Des}(I)}{\bf F}_{\sigma}$ span a sub $*$-algebra of ${\bf FQSym}$ isomorphic to $({\bf Sym},*)$, which is therefore anti-isomorphic to the Solomon descent algebra. **

3. Transformations of alphabets

3.1. Transformations in ${\it QSym}$

Recall that the classical Cauchy identities for symmetric functions can be extended to the dual pair of Hopf algebras $(QSym,{\bf Sym})$ as follows. Let $X$ be a totally ordered alphabet of commutative variables, and $A$ be an alphabet of noncommuting variables, also totally ordered. The product alphabet $XA$ is the set of products $xa$ endowed with the lexicographic order on the pairs $(x,a)$. We can thus define the noncommutative symmetric functions of $XA$ and we have

[TABLE]

for any pair $(U,V)$ of mutually dual bases [10] (the arrows mean that the products are to be taken in increasing order).

We can introduce a second commutative alphabet $T$, and compute in two ways

[TABLE]

The alphabet $T$ denoted by $\frac{1}{1-t}$ is $\{t^{n}|n\geq 0\}$, ordered by $t^{i}<t^{j}$ iff $i>j$ (which would be true for numerical values of $t$ such that the geometric series converge). We introduce the notations

[TABLE]

The maps

[TABLE]

are algebra automorphisms, and their inverses are consistently denoted by

[TABLE]

Here, $(1-t)$ is an example of a virtual alphabet. More generally, a virtual alphabet $T$ is defined as a morphism of algebras $\chi_{T}$

[TABLE]

from $QSym$ to some commutative algebra. This defines the $TA$ transform in ${\bf Sym}$ by

[TABLE]

and by duality, the $XT$ transform on $QSym$

[TABLE]

where

[TABLE]

The $(t-1)$ transform is defined by writing $t-1=t(1-t^{-1})$ so that $M_{I}(X(t-1))=t^{|I|}M_{I}(X(1-t^{-1}))$.

We note for further reference the specializations

[TABLE]

3.2. Transformations in ${\bf WQSym}$

The $1/(1-t)$ transform may be extended from ${\it QSym}$ to ${\bf WQSym}$ by setting

[TABLE]

endowed with the total order $a_{i}t^{j}<a_{k}t^{l}\Leftrightarrow i<k$ or $i=k$ and $j>l$. Then, the commutative image of ${\bf M}_{u}\left(\frac{A|}{|1-t}\right)$ is $M_{I}\left(\frac{X|}{|1-t}\right)$, where $I={\rm ev}(u)$.

The inverse transformations are consistently denoted by ${\bf M}_{u}\mapsto{\bf M}_{u}(A(1-t))$ on ${\bf WQSym}$ and $S^{I}\mapsto S^{I}((1-t)A)$ on ${\bf Sym}$. These have been investigated in [23, 18].

The adjoint map of ${\bf M}_{u}\mapsto{\bf M}_{u}(A(1-t))$ is ${\bf N}_{u}\mapsto{\bf N}_{u}*\sigma_{1}((1-t)A)$, and similarly for the inverse maps.

Although there is no known polynomial realization of ${\bf WQSym}^{*}$, it will be convenient to define ${\bf N}_{u}(TA)$ as

[TABLE]

Let

[TABLE]

Proposition 3.1**.**

Let $u$ and $w$ be two packed words of the same size. Let $w^{(i)}=\operatorname{pack}(w_{j_{1}}w_{j_{2}}\cdots w_{j_{p}})$, where $\{j_{1},\ldots,j_{p}\}=\{j|u_{j}=i\}$. Then,

[TABLE]

*Proof – *Since the packing process commutes with the right action of the symmetric group (see (16)), we can apply to $u$ the smallest permutation $\sigma$ such that $u\sigma$ is nondecreasing (i.e., $\sigma={\rm std}(u)^{-1}$), so that $\operatorname{pack}{\binom{u\sigma}{v\sigma}}=w\sigma$. We can therefore assume that $u$ is nondecreasing. First note that no relation is required between the letters of $v$ corresponding to different letters of $u$. The only order constraints are among places where $u$ has identical letters, and these are the same as in the corresponding letters of $w$. This is precisely the definition of the convolution on packed words, describing the product of the ${\bf M}$ basis [22].

Thus,

[TABLE]

and by duality,

[TABLE]

The morphism $\chi_{T}$ defining a virtual alphabet is naturally extended to ${\bf WQSym}$ by setting ${\bf M}_{u}(T)=M_{I}(T)$, where $I={\rm ev}(u)$.

A packed word $v$ is said to refine $u$ if for all $i<j$, $v_{i}>v_{j}\Longleftrightarrow u_{i}\geq u_{j}$ and $v_{i}=v_{j}\Longrightarrow u_{i}=u_{j}$. In this case, we write $v\geqslant_{\rm ref}u$. This is the usual notion of refinement on set compositions: each block of $u$ is a union of consecutive blocks of $v$.

For example, the packed words finer than $212$ are $212$, $213$, and $312$. The packed words finer than $2122$ are

[TABLE]

Lemma 3.2**.**

The coefficient of ${\bf M}_{v}(A)$ in (44) is 0 if $u$ is not finer than $v$, and equal to the coefficient of $M_{{\rm ev}(u)}(X)$ in $M_{{\rm ev}(v)}(XT)$ otherwise.

*Proof – *By definition, the words $u^{(i)}$ exist only when $u$ is finer than $v$, and then, ${\bf M}_{w^{(1)}}{\bf M}_{w^{(2)}}\cdots{\bf M}_{w^{(\max(u))}})(T)$ is equal to $M_{I_{1}}(T)\cdots M_{I_{s}}(T)$, where $I={\rm ev}(v)$, $J={\rm ev}(u)$ and $I_{k}\vDash j_{k}$ for all $k$.

4. Dyck graphs

Definition 4.1**.**

A Dyck graph is a simple undirected graph $G$ with vertex set $V(G)=[n]$ and edge set $E(G)$ represented as pairs $(i<j)$ such that if $(i,j)\in E(G)$, then $(i^{\prime},j^{\prime})\in E(G)$ for all $i\leq i^{\prime}<j^{\prime}\leq j$.

Define for $\sigma\in{\mathfrak{S}}_{n}$

[TABLE]

We shall make use of descent bottoms of a permutation associated with a graph, that are the values $\sigma_{i+1}$ such that $\sigma_{i}>\sigma_{i+1}$ and $(\sigma_{i+1},\sigma_{i})\not\in E(G)$. For example, if $G$ is the graph

[TABLE]

(labelled 1–5 from left to right), and $\sigma=35142$, then ${\rm inv}_{G}(\sigma)=\{(2,3),(4,5)\}$, $\operatorname{Des}_{G}(\sigma)=\{2,4\}$, ${\rm maj}_{G}(\sigma)=6$ and the descent bottoms of $\sigma$ are $\{1,2\}$.

Set ${\rm st}_{G}(\sigma)={\rm inv}_{G}(\sigma)+{\rm maj}_{G}(\sigma)$ [24, 17]. Recall the notation $[n]_{t}=1+t+\cdots t^{n-1}$.

Theorem 4.2**.**

Let $G$ be a Dyck graph. For any $\sigma\in{\mathfrak{S}}_{n-1}$,

[TABLE]

where $H$ is the restriction of $G$ to the interval $[1,n-1]$ and $\sigma\,{\mathchoice{\mathbin{\hbox{\vbox{ \hbox{\vrule\hskip 3.5pt\vrule height=7.0pt,width=0.0pt}\hrule}\vbox{ \hbox{\vrule\hskip 3.5pt\vrule height=7.0pt,width=0.0pt\vrule}\hrule}}}}{\mathbin{\hbox{\vbox{ \hbox{\vrule\hskip 3.0pt\vrule height=6.0pt,width=0.0pt}\hrule}\vbox{ \hbox{\vrule\hskip 3.0pt\vrule height=6.0pt,width=0.0pt\vrule}\hrule}}}}{\mathbin{\hbox{\vbox{ \hbox{\vrule\hskip 2.0pt\vrule height=4.0pt,width=0.0pt}\hrule}\vbox{ \hbox{\vrule\hskip 2.0pt\vrule height=4.0pt,width=0.0pt\vrule}\hrule}}}}{\mathbin{\hbox{\vbox{ \hbox{\vrule\hskip 1.5pt\vrule height=3.0pt,width=0.0pt}\hrule}\vbox{ \hbox{\vrule\hskip 1.5pt\vrule height=3.0pt,width=0.0pt\vrule}\hrule}}}}}\,n$ means the set of all words $\tau$ such that the restriction of $\tau$ to $[1,n-1]$ gives back $\sigma$.

Corollary 4.3**.**

Let $\sigma|_{[1,k]}$ denote the restriction of $\sigma$ to the interval $[1,k]$. The map

[TABLE]

is a bijection from ${\mathfrak{S}}_{n}$ to the set of integer vectors $v\in{\mathbb{N}}^{n}$ such that $v_{i}\leq n-i$.

Thus, $c$ is a code which interpolates between the Lehmer code (complete graph) and the majcode (no edges).

In particular, we recover a particular case of a result of Kasraoui [17]:

Corollary 4.4**.**

For any Dyck graph $G$, the statistic ${\rm st}_{G}$ is Mahonian:

[TABLE]

The previous theorem is a direct consequence of the following lemma.

Lemma 4.5**.**

Consider the permutations $\tau$ obtained from $\sigma$ by inserting $n$ at each of the $n$ possible positions.

Then, ${\rm st}_{G}(\tau)-{\rm st}_{H}(\sigma)$ takes all the values from [math] to $n-1$ in this order if one visits the insertion positions in the following order:

start with the rightmost position, then, from right to left, insert $n$ to the left of the values $k$ such that $(k,n)\in E$ or that are descent bottoms of $\sigma$, then from left to right, run through the remaining ones.

*Proof – *First note that (51) implies that ${\rm st}_{G}$ is Mahonian by induction on $n$.

Let us now prove it. Let $H$ be the restriction of $G$ to $[n-1]$. There are four cases to be distinguished according to whether $\tau$ is obtained by inserting $n$:

1. (1)

at the end of $\sigma$: then ${\rm st}_{G}(\tau)={\rm st}_{H}(\sigma)$. 2. (2)

to the left of a $k$ such that $(k,n)\in E(G)$. Then, $(k,k^{\prime})\in E(G)$ for all $k<k^{\prime}<n$, so that $k$ cannot be a descent bottom of $\sigma$. Thus,

[TABLE]

where $d_{H}(k)$ is the number of $H$-descent bottoms of $\sigma$ to the right of $n$ (since each descent is shifted by one position to the right), and $e_{G}(k)$ is the number of $k^{\prime}$ to the right of $n$ such that $(k^{\prime},n)\in E(G)$ (since all these values have an inversion with $n$). 3. (3)

to the left of an $H$-descent bottom $k$. Then, the letter $\ell$ preceding $k$ in $\sigma$ is such that $(k,\ell)\not\in E(G)$, so that $(l,k)\not\in E(G)$. Therefore, inserting $n$ between $\ell$ and $k$ creates a descent $(n,k)$ in $\tau$ which takes the place of the descent $(k,i)$ in $\sigma$ just one position to the right. Thus,

[TABLE]

as in the previous case. Here, the $+1$ due to moving the descent bottom $k$ one place to the right is taken into account in $d_{H}(k)$. 4. (4)

to the left of a $k$ such that $(k,n)\not\in E(G)$ and $k$ is not an $H$-descent bottom. Then $(n,k)$ is a new descent and

[TABLE]

Let us now consider the sequence of insertions described in Lemma 4.5.

In the first part of the sequence going from right to left, one easily sees that the values of $d_{H}(k)+e_{G}(k)$ increase by one at each step since we stop at each element either creating an inversion with $n$ or being a descent bottom. In the second part moving from left to right, the same property holds: at each step, the value of $\sigma^{-1}_{k}+d_{H}(k)+e_{G}(k)$ increases by one since, between two elements, $\sigma^{-1}_{k}$ changes by one plus the number of values between these that are either descent bottoms or related with $n$ in $G$, which is compensated by the fact that $d_{H}$ and $e_{G}$ decrease respectively on descent bottoms or values connected to $n$ in $G$.

Finally, it is easily checked that both ways of evaluating the increment of ${\rm st}_{H}$ corrresponding to the leftmost insertion position do agree, whence the claim.

Note 4.6**.**

This argument is similar to the one used for the maj-code (see, e.g., [16]). In particular, the definition of the sequence going backward then forward to visit each possible insertion position of $n$ is essentially the same: one just has to add a special case when $(k,n)\in E(G)$. **

Note 4.7**.**

The statistic $s_{G}(\sigma)$ interpolates in a Catalan number of ways between the inversions number ($G$ is the complete graph) and the major index ($G$ is the graph with no edges). **

Example 4.8**.**

Consider the graph

on $V(G)=[6]$ with $E(G)=\{12,23,24,34,45,46,56\}$ and the permutation $\sigma=52314$.

Then $4$ and $5$ belong to case (2), $1$ and $2$ belong to case (3), and $3$ belongs to case (4). Then the order is ${}^{4}5^{3}2^{5}3^{2}1^{1}4^{0}$, where the exponents encode the sequence. **

Proposition 4.9**.**

For a Dyck graph,

[TABLE]

*Proof – *According to [24, Theorem 9.3], the principal specialization of ${\rm X}_{G}$ satisfies

[TABLE]

where $(q;q)_{n}=(1-q)(1-q^{2}\textdegree\cdots(1-q^{n})$. For $q=t$, ${\rm X}_{G}$ being symmetric, this yields

[TABLE]

where the first equality comes from the fact that for a symmetric function $f$ homogeneous of degree $n$, $f(-X)=(-1)^{n}\omega f(X)$. Dividing by $[n]_{t}!$, we are left with

[TABLE]

5. The Guay-Paquet Hopf algebra

In his proof of the Shareshian-Wachs conjecture [12], Guay-Paquet introduces a Hopf algebra ${\mathcal{G}}$ based on ordered graphs, depending on a parameter $t$, and such that the map sending a graph to itsromatic quasi-symmetric function is a morphism of Hopf algebras ${\mathcal{G}}\rightarrow QSym$.

Its basis consists of finite simple undirected graphs with vertices labelled by the integers from 1 to $n=|V(G)|$. The product is the shifted concatenation: $G\cdot H=G\cup H[n]$ where $H[n]$ is $H$ with labels shifted by the number $n$ of vertices of $G$.

The parameter $t$ arises in the coproduct. If $G$ is a graph on $n$ vertices and $w\in[r]^{n}$, regarded as a coloring of $G$, we denote by $G|_{w}$ the tensor product $G_{1}\otimes\cdots\otimes G_{r}$ of the restrictions of $G$ to vertices colored $1,2,\ldots,r$. The $r$-fold coproduct is then

[TABLE]

At $t=1$, ${\mathcal{G}}$ becomes cocommutative and is isomorphic to an algebra introduced in [25].

It is also proved in [12] that the subspace ${\mathcal{D}}$ of ${\mathcal{G}}$ spanned by Dyck graphs is a Hopf subalgebra. At $t=1$, it is a free cocommutative graded connected Hopf algebra of graded dimension Catalan, and is therefore isomorphic to ${\bf CQSym}$ [15].

From these properties, we obtain a simple conceptual proof of (1):

Proposition 5.1**.**

[TABLE]

*Proof – *If $G$ is a Dyck graph and $u$ a packed word, denote by $G_{i}(u)$ the restriction of $G$ to the vertices $j$ such that $u_{j}=i$. Then the coefficient of $M_{I}(X)$ in ${\rm X}_{G}\left(t,\frac{X|}{|t-1}\right)$ is

[TABLE]

Dualizing the product $S^{I}=S_{i_{1}}\cdots S_{i_{r}}$, this is equal to

[TABLE]

and since $G\mapsto{\rm X}_{G}$ is a morphism of Hopf algebras, the iterated coproduct can be evaluated by (57), which yields

[TABLE]

by Prop. 4.9.

Thanks to Lemma 3.2, this argument can be extended to the noncommutative case.

6. The noncommutative chromatic quasi-symmetric function

6.1. A noncommutative analogue of ${\rm X}_{G}$

Given a Dyck graph $G$, define

[TABLE]

For example,

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Proposition 6.1**.**

$G\mapsto{\bf X}_{G}(A)$* is a morphism of Hopf algebras from ${\mathcal{G}}$ to ${\bf WQSym}$.*

*Proof – *The argument is essentially the same as for $QSym$. Multiplicativity is clear:

[TABLE]

Next, the coefficient of ${\bf M}_{u_{1}}\otimes{\bf M}_{u_{2}}$ in $\Delta{\bf X}_{G}$ is nonzero if and only if $u_{1}\in\rm PC\,(G_{1})$ and $u_{2}\in\rm PC\,(G_{2})$ for some splitting of the vertices of $G$ into two complementary subsets, detemined by a word $w\in\{1,2\}^{n}$ as in (57).

Each such splitting determines a proper coloring $u$ of $G$: color the vertices of $G_{1}$ with $u_{1}$ and those of $G_{2}$ with the shifted word $u_{2}[\max(u_{1})]$ (recall that the shifted word $u[k]$ is obtained by adding $k$ to all values of $u$). Thus, $u_{1}$ and $u_{2}$ are the restrictions of $u$ to two consecutive intervals, so that ${\bf M}_{u_{1}}\otimes{\bf M}_{u_{2}}$ occurs in $\Delta{\bf M}_{u}$. In particular, note that $u$ belongs to the shifted shuffle of $u_{1}$ and $u_{2}$.

Conversely, any $u\in\rm PC\,(G)$ and any term ${\bf M}_{u_{1}}\otimes{\bf M}_{u_{2}}$ occuring in $\Delta{\bf M}_{u}$ uniquely determines a splitting of $V(G)$ into two complementary subsets: the vertices of $G_{1}$ correspond to the positions of the letters of the subword $u_{1}$ of $u$. This proves that at $t=1$, ${\bf X}$ is indeed a morphism of coalgebras.

Now, in the above situation, we have

[TABLE]

where $r$ is the number of edges $(i<j)$ of $G$ with $u_{i}<u_{j}$ which are neither in $G_{1}$ nor in $G_{2}$. These correspond precisely to the $G$-ascents of the word $w\in\{1,2\}^{n}$ determining the splitting.

From the product rule of the ${\bf M}$ basis of ${\bf WQSym}$, one can easily check that

[TABLE]

and that

[TABLE]

For the coproduct, one can check the following example:

[TABLE]

This example also allows to check that the restriction of the coproduct to the subalgebra of Dyck graphs is cocommutative.

6.2. Noncommutative unicellular LLT polynomials

Theorem 6.2**.**

[TABLE]

The r.h.s. is therefore a noncommutative lift of the LLT polynomial ${\rm LLT}_{G}$.

*Proof – *The coefficient of ${\bf M}_{v}$ in ${\bf X}_{G}\left(t,\frac{A|}{|t-1}\right)$ is

[TABLE]

Up to a power of $t$, the sum in the right-hand side of the bracket is the product of the quasi-symmetric chromatic polynomials of the graphs $G_{i}(v)$ evaluated at $\frac{X|}{|t-1}$. The power of $t$ corresponds to the $G$-ascents of $v$ on the deleted edges, that is

[TABLE]

Definition 6.3**.**

Given a Dyck graph $G$, the non-commutative LLT polynomial ${\bf LLT}_{G}$ is

[TABLE]

Note 6.4**.**

Alternatively, the r.h.s of (79) can be interpreted as a duality bracket for the pair $({\bf WQSym}^{*},{\bf WQSym})$:

[TABLE]

and evaluating the iterated coproducts $\Delta^{r}{\bf M}_{u}$ leads to the same conclusion. **

6.3. Special case: path graphs

Let $G_{n}$ be the graph on $[n]$ with edges $(i,i+1)$. Then,

[TABLE]

If we embed ${\bf Sym}$ in ${\bf WQSym}$ by sending $S_{n}$ to the sum of nonincreasing words

[TABLE]

we can write

[TABLE]

so that

[TABLE]

which gives back by commutative image the generating series of [24, C.2]. The images of $S_{n}$ and $\Lambda_{n}$ by the $A\mapsto A(t-1)$ transform are given by

[TABLE]

and its inverse ($\lambda_{-t}:=(\sigma_{t})^{-1}$)

[TABLE]

Hence,

[TABLE]

At $t=1$, this gives back the well-known fact that the sum of all Smirnov words is the inverse of the alternating sum of constant words.

7. The Dyck graphs subalgebra of ${\bf WQSym}$

The goal of this section is to prove

Theorem 7.1**.**

The restriction of the morphism of Hopf algebras $G\mapsto{\bf X}_{G}(t,A)$ from ${\mathcal{G}}$ to ${\bf WQSym}$ to the subalgebra ${\mathcal{D}}$ of Dyck graphs is injective.

We shall prove that the images of the Dyck graphs are already linearly independent for $t=1$.

7.1. The Hopf algebra ${\bf WSym}$

The ${\bf X}_{G}(1,A)$ are the noncommutative chromatic polynomials defined by Gebhard [8, 9], and thus belong to the algebra of symmetric functions in noncommuting variables $a_{i}$, denoted here by ${\bf WSym}$.

It consists of the invariants of ${\mathfrak{S}}(A)$ acting by automorphisms on the free algebra ${\bf K}\langle A\rangle$. Two words $u=u_{1}\cdots u_{n}$ and $v=v_{1}\cdots v_{n}$ are in the same orbit whenever $u_{i}=u_{j}\Leftrightarrow v_{i}=v_{j}$. Thus, orbits are parametrized by set partitions into at most $|A|$ blocks. Assuming that $A$ is infinite, we obtain an algebra based on all set partitions whose monomial basis is defined by

[TABLE]

where $O_{\pi}$ is the set of words such that $w_{i}=w_{j}$ iff $i$ and $j$ are in the same block of $\pi$.

As an example of expansion of a chromatic polynomial in terms of the ${\bf m}$, we have

[TABLE]

The product of the ${\bf m}$ is given by the rule

[TABLE]

where $E(\pi^{\prime},\pi^{\prime\prime})$ consists of all set partitions whose parts are either a part of $\pi^{\prime}$, a part of $\pi^{\prime\prime}$, or a union of a part of $\pi^{\prime}$ and a part of $\pi^{\prime\prime}$.

Since set partitions are equivalence classes of set compositions which are in bijection with packed words, we shall often denote a set partition as the lexicographically minimal packed word in its class, which amounts to representing a set partition by the set composition obtained by ordering its blocks w.r.t. their minima. For example, $\{\{1,4\}\{2,5\},\{3\}\}$ will be represented by $12312$.

Let us illustrate this notation on two examples of the product:

[TABLE]

and

[TABLE]

7.2. The chromatic polynomials

To prove the linear independence of the images of the Dyck graphs, we shall show that they are triangular with respect to a basis of a subalgebra of ${\bf WSym}$ based on nonnesting partitions.

Define the denesting ${\rm dn}(\pi)$ of a set partition $\pi$ as the nonnesting partition $\pi^{\prime}$ obtained by iterating the following process: for each sequence $i<j<k<l$ such that $j$ and $k$ are in a block $B_{1}$ of $\pi$ and $i$ and $l$ in another block $B_{2}$ containing no intermediate value $i<r<l$, i.e., $B_{2}=\{m_{1}<\cdots<m_{p}=i<m_{p+1}=l<\cdots m_{r}\}$, split $B_{2}$ into $B^{\prime}_{2}=\{m_{1},\ldots,i\}$ and $B^{\prime\prime}_{2}=\{l,\ldots<m_{r}\}$.

Up to $n=3$, all set partitions are fixed by the denesting algorithm and there is only one set partition $\pi$ of size $4$ such that ${\rm dn}(\pi)\not=\pi$. In terms of set partitions, it is $\{\{1,4\},\{2,3\}\}$ and ${\rm dn}(\pi)=\{\{1\},\{2,3\},\{4\}\}$. In terms of packed words, it is $1221$ and ${\rm dn}(1221)=1223$.

If $\pi=12341312$, then ${\rm dn}(\pi)=12341356$.

Proposition 7.2**.**

For a nonnesting partition $\pi$, define

[TABLE]

Then, the ${\bf\tilde{m}}_{\pi}$ form the basis of a subalgebra of ${\bf WSym}$ of homogeneous dimensions given by the Catalan numbers.

For example,

[TABLE]

[TABLE]

[TABLE]

[TABLE]

*Proof – *Since the product of the ${\bf m}$-basis is multiplicity-free, we just have to check that for any partition $\pi^{\prime}$, ${\bf m}_{\pi^{\prime}}$ occurs in ${\bf\tilde{m}}_{\pi_{1}}{\bf\tilde{m}}_{\pi_{2}}$ if and only if ${\bf m}_{{\rm dn}(\pi^{\prime})}$ occurs in this product.

If $\pi_{1}\vdash[n_{1}]$ and $\pi_{2}\vdash[n_{2}]$, then ${\bf m}_{\pi^{\prime}}$ occurs in ${\bf\tilde{m}}_{\pi_{1}}{\bf\tilde{m}}_{\pi_{2}}$ if and only if ${\rm dn}(\pi^{\prime}|_{[1,n_{1}]})=\pi_{1}$ and ${\rm dn}(\pi^{\prime}|_{[n_{1},n_{1}+n_{2}]})=\pi_{2}$.

Since the denesting process is obviously compatible with restriction to intervals, this is equivalent to ${\rm dn}(\pi^{\prime})|_{[1,n_{1}]}=\pi_{1}$ and ${\rm dn}(\pi^{\prime})|_{[n_{1},n_{1}+n_{2}]}=\pi_{2}$, which is the condition for ${\bf m}_{{\rm dn}(\pi^{\prime})}$ to occur in ${\bf\tilde{m}}_{\pi_{1}}{\bf\tilde{m}}_{\pi_{2}}$.

Lemma 7.3**.**

For a Dyck graph $G$,

[TABLE]

where the sum runs over nonnesting partitions, and $\pi(i)$ denotes the block containing $i$.

*Proof – *If ${\bf m}_{\pi^{\prime}}$ occurs in ${\bf X}_{G}$, then so does ${\bf\tilde{m}}_{{\rm dn}(\pi^{\prime})}$, since ${\rm dn}(\pi^{\prime})$ is finer than $\pi^{\prime}$, so is associated with proper colorings as well.

Conversely, if ${\bf m}_{\pi^{\prime}}$ does not occur in ${\bf X}_{G}$, there exist $i,j$ with $|j-i|$ minimal such that $(i,j)\in E(G)$ and $i,j$ in the same block of $\pi^{\prime}$. By minimality of $|j-i|$, $i$ and $j$ are consecutive in their block. Moreover, $(i,j)\in E(G)$ implies that $(i^{\prime},j^{\prime})\in E(G)$ for all $i<i^{\prime}<j^{\prime}<j$. Still by minimality of $|j-i|$, $i+1,\ldots,j-1$ are all in different blocks. Hence, $i$ and $j$ would not be separated by the denesting process, so that ${\bf m}_{{\rm dn}(\pi^{\prime})}$ does not occur in ${\bf X}_{G}$ either.

There is a simple bijection $\eta$ between nonnesting partitions $\pi$ (represented as diagrams of arcs) and Young diagrams $\lambda$ contained in the staircase partition $(n-1,\ldots,2,1)$, represented as sets of cells above the diagonal in an $n\times n$ square: the arcs of $\pi$ are the coordinates of the corners of $\lambda$. For example, the partition $\lambda=(221)$ corresponds to the nonnesting partition $i\{1,3\},\{2,4\},\{5\}\}$ which is read on the coordinates of the corners of the diagram. The edges of corresponding graph $G$ are the coordinates of the empty cells above the diagonal, $(1,2),(2,3),(3,4),(3,5),(4,5)$.

[TABLE]

Thanks to that bijection, there is a natural partial order on nonnesting partitions: the Young lattice restricted to partitions contained in the staircase. We shall say that $\pi^{\prime}\leq\pi$ if the image of $\pi^{\prime}$ is included in the image of $\pi$.

Lemma 7.4**.**

Given a Dyck graph $G$,

[TABLE]

where the sum runs over nonnesting partitions smaller than the nonnesting partition $\pi_{G}$ corresponding to the Young diagram encoding $G$.

*Proof – *Let $\lambda=\eta(\pi_{G})$. Let $\pi^{\prime}$ be a nonnesting set partition. If $\eta(\pi^{\prime})\subseteq\lambda$ then, thanks to the bijection between partitions and Dyck graphs, for all $(i,j)\in E(G)$, all pairs $(i^{\prime},j^{\prime})$ such that $i\leq i^{\prime}<j^{\prime}\leq j$ are also edges of $G$, so that $i^{\prime}$ and $j^{\prime}$ can never be in the same part of $\pi$, hence of $\pi^{\prime}$. So thanks to the previous lemma, ${\bf\tilde{m}}_{\pi^{\prime}}$ appears in the expansion of ${\bf X}_{G}$.

Conversely, if $\eta(\pi^{\prime})\not\subseteq\lambda$, then there exists a corner $(i,j)$ of $\eta(\pi^{\prime})$ that does not belong to $\lambda$. Then $(i,j)$ is an edge of $G$ since it is an empty cell in $\lambda$ but $i$ and $j$ are in the same block of $\pi^{\prime}$ since they are consecutive by definition. So ${\bf\tilde{m}}_{\pi^{\prime}}$ does not appear in the expansion of ${\bf X}_{G}$.

For example,

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

8. A multiplicative basis

Recall that the reverse refinement order, denoted by $\leq$, on compositions is such that $I=(i_{1},\ldots,i_{k})\geq J=(j_{1},\ldots,j_{l})$ iff $\{i_{1},i_{1}+i_{2},\ldots,i_{1}+\cdots+i_{k}\}$ contains $\{j_{1},j_{1}+j_{2},\ldots,j_{1}+\cdots+j_{l}\}$. In this case, we say that $I$ is finer than $J$. For example, $(2,1,2,3,1,2)\geq(3,2,6)$.

This order can be extended to packed words as follows. To avoid confusion with the order of packed words defined previously, we say that $w$ is strongly finer than $w^{\prime}$, and write $w\geq w^{\prime}$, iff $w$ and $w^{\prime}$ have same standardized word and the evaluation of $w$ is finer than the evaluation of $w^{\prime}$. It also amounts to asking that $w$ and $w^{\prime}$ have same standardized word and that $w\geqslant_{\rm ref}w^{\prime}$.

For example, the packed words strongly finer than $212$ are $212$ and $213$. The packed words strongly finer than $2122$ are

[TABLE]

to be compared with the packed words finer that $2122$ shown in (45).

Given a permutation, let us define the set $A(\sigma)$ of the advances of $\sigma$ as the set of values $i$ such that $i+1$ is to the right of $i$ in $\sigma$. Note that it is the complementary set over $[1,n-1]$ of the usual recoils. Let ${\rm DST\,}(\sigma)$ be the set of packed words of standardization $\sigma$.

Lemma 8.1**.**

Let $\sigma$ be a permutation. Then the elements of ${\rm DST\,}(\sigma)$ are in bijection with the subsets of $A(\sigma)$.

In particular, this set of words has a natural structure of boolean lattice.

*Proof – *Let $i$ be an element of $A(\sigma)$ and let $j<k$ be the respective positions of $i$ and $i+1$ in $\sigma$. Then in any element $w$ of ${\rm DST\,}(\sigma)$, either $u_{j}=u_{k}$ or $u_{k}=u_{j}+1$. Since there are two independent choices for all elements of $A(\sigma)$, the result holds.

The inverse map going from ${\rm DST\,}(\sigma)$ to subsets of $A(\sigma)$ is given by the rule: put $i$ in its corresponding subset of $A(\sigma)$ if $u_{k}=u_{j}+1$.

Corollary 8.2**.**

Let $w$ be a word. Then the elements strongly coarser than $w$ are an interval of the boolean order of ${\rm DST\,}(w)$ described in Lemma 8.1, hence themselves a boolean order consisting of the subsets of $A({\rm std}(w))$ that, following the notations of the previous lemma, necessarily contains the elements $i$ such that $w_{j}=w_{k}$.

The boolean order of the packed words of standardized $13425$ is given in Figure 1.

Let [22, 2]

[TABLE]

For example,

[TABLE]

[TABLE]

Since $(\Phi_{u})$ is triangular over $({\bf M}_{u})$, it is a basis of ${\bf WQSym}$. Note that the order used for summation is a restriction of the refinement order on compositions, so is a boolean lattice. Hence,

[TABLE]

For example,

[TABLE]

By construction, the basis $\Phi$ satisfies a product formula similar to that of Gessel’s basis $F_{I}$ of ${\it QSym}$ (whence the choice of notation). We shall not state it since we will not need it in the sequel but here follows an example illustrating the similarity with Gessel’s basis.

[TABLE]

Proposition 8.3**.**

The noncommutative $t$-chromatic function is $\Phi$-positive:

[TABLE]

where $\min_{G}(\sigma)$ is the packed word $u$ defined as follows: let

[TABLE]

Then,

[TABLE]

All non trivial examples (excluding the case of the complete graph where $S$ is always empty) of size $3$ are given below.

First, here are all sets $S$ and then all packed words $\min_{G}(\sigma)$.

[TABLE]

[TABLE]

We then deduce

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

*Proof – *Since $\Phi_{w}$ expanded in the ${\bf M}$ basis is a sum over elements of ${\rm DST\,}({\rm std}(w))$, we have to show two things: first, the monomials in $t$ that are coefficients of the ${\bf M}$ are constant among elements of ${\rm DST\,}({\rm std}(w))$ and among the elements of ${\rm DST\,}({\rm std}(w))$, the elements appearing in ${\bf X}_{G}$ expanded in the ${\bf M}$ basis are exactly the elements finer than $\min_{G}(\sigma)$.

Concerning the coefficients (monomials in $t$) of these elements, if $w$ appears with a coefficient $t^{i}$ then this is also the coefficient of ${\rm std}(w)$ and more generally of any element strongly finer than $w$. Indeed, the power of $t$ counts the ascents among the pairs of edges of $G$ and that does not change from $w$ to $w^{\prime}\geq w$ if $w$ appears in ${\bf X}_{G}$. This comes from the fact that all ascents of $w$ are ascents of $w^{\prime}$ and that the only positions $(i,j)$ that could add an ascent from $w$ to $w^{\prime}$ are those such that $w^{\prime}_{i}>w^{\prime}_{j}$ and $w_{i}=w_{j}$, but in that case $(i,j)$ cannot be an edge of $G$ since $w$ is a proper coloring of $G$, and hence cannot count as an ascent of $w^{\prime}$.

Let us now show that the packed words with standardized $\sigma$ appearing in ${\bf X}_{G}$ are the elements finer than $\min_{G}(\sigma)$. Recall that thanks to Lemma 8.1, all words having a given standardized $\sigma$ form a boolean lattice when equipped with the strong refinement order, and that any element corresponds to a subset of the set of values $i$ such that $i-1$ is to the left of $i$ in $\sigma$ (or, equivalently $\sigma^{-1}_{i-1}<\sigma^{-1}_{i}$).

Since we are looking for the packed words that are proper colorings of $G$, it is pretty clear that the subsets containing an $i$ such that $(\sigma^{-1}_{i-1},\sigma^{-1}_{i})\in E(G)$ cannot bring proper colorings since two connected vertices would have the same color. Conversely, excluding those values necessarily brings a proper coloring.

So all packed words appearing in the expansion of ${\bf X}_{G}$ in the $M$ basis with a given standardized word $\sigma$ are strongly finer than $\min_{G}(\sigma)$ and have all same coefficient, whence the statement.

Corollary 8.4** ([24], Thm 3.1).**

The chromatic quasi-symmetric function of a Dyck graph is $F$-positive and its expansion is

[TABLE]

where ${\rm inv}_{G}(\sigma)$ is the pairs $(i<j)$ such that $(\sigma_{i},\sigma_{j})\in E(G)$ and $\sigma_{i}>\sigma_{j}$, and $DES_{P}(\sigma)$ is the composition encoding the set of $i$ such that $(\sigma_{i},\sigma_{i+1})\not\in E(G)$ and $\sigma_{i}>\sigma_{i+1}$, and where $\tilde{}$ denotes the conjugate composition.

*Proof – *Our formula in ${\bf WQSym}$ is projected to this expression by the morphism sending $\Phi_{w}$ to $F_{{\rm ev}(w)}$: the contribution of $\sigma$ in our equation is the contribution of $\sigma^{\prime}={\rm inv}(r(\sigma))$ in their equation, where $r(\sigma)$ sends each value $i$ to $n+1-i$ if $\sigma\in{\mathfrak{S}}_{n}$.

Indeed, our ascents of a permutation go to the inversions of [24] through ${\rm inv}\circ r$ since $r$ changes ascents to inversions and ${\rm inv}$ exchanges values and positions. Moreover, two values $i$ and $i+1$ of $\sigma$ are equal in $\min_{G}(\sigma)$ (hence are not a descent of the composition ${\rm ev}(\min_{G}(\sigma))$) iff they are increasing and their positions do not correspond to an edge of $G$. Since $\sigma^{\prime}$ can also be described as $\sigma^{\prime}={\overline{{\rm inv}(\sigma)}}$, where $\bar{w}$ denotes the mirror-image of $w$, this exactly translates in $\sigma^{\prime}$ as the values in positions $n-i$, $n+1-i$ that decrease and do not form an edge of $G$, which is exactly the definition of the conjugate of $DES_{P}(\sigma^{\prime})$.

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}{}\pgfsys@moveto{2.35693pt}{0.0pt}\pgfsys@curveto{2.35693pt}{1.30171pt}{1.30171pt}{2.35693pt}{0.0pt}{2.35693pt}\pgfsys@curveto{-1.30171pt}{2.35693pt}{-2.35693pt}{1.30171pt}{-2.35693pt}{0.0pt}\pgfsys@curveto{-2.35693pt}{-1.30171pt}{-1.30171pt}{-2.35693pt}{0.0pt}{-2.35693pt}\pgfsys@curveto{1.30171pt}{-2.35693pt}{2.35693pt}{-1.30171pt}{2.35693pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{0.5}{0.0}{0.0}{0.5}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ 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{{}}{}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}{{}}{{{{}}{{}}{{}}{{}}}{{{{}}{}{}{}{}{{}}}} }{{{{}}{{}}{{}}{{}}{{}}}{{{{}}{}{}{}{}{{}}}} }{{}{}}{{}} {}{}{}{{{}}{{}}{{}}} {{{}}{{}}{{}}} {}{{}}{}{{}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }{}\pgfsys@moveto{15.45483pt}{2.12776pt}\pgfsys@curveto{20.5236pt}{10.90717pt}{36.38191pt}{10.90717pt}{41.45068pt}{2.12776pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}{{}}{{{{}}{{}}{{}}{{}}}{{{{}}{}{}{}{}{{}}}} }{{{{}}{{}}{{}}{{}}{{}}}{{{{}}{}{}{}{}{{}}}} }{{}{}}{{}} {}{}{}{{{}}{{}}{{}}} {{{}}{{}}{{}}} {}{{}}{}{{}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }{}\pgfsys@moveto{43.9076pt}{2.12776pt}\pgfsys@curveto{48.97636pt}{10.90717pt}{64.83467pt}{10.90717pt}{69.90344pt}{2.12776pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ and the permutation $314652$ contribute in our case to a term $t^{3}{\bf M}_{212321}$, whereas $G$ and $(\overline{314652})^{-1}=(453162)^{-1}=453162$ contribute in the case of [24] as $t^{3}M_{231}$.

8.1. Noncommutative LLT polynomials

To formulate the noncommutative analogue of the $F$-expansion of unicellular LLT polynomials, we need another lift of the $F$-basis, defined as

[TABLE]

where the bar involution sends a word to its mirror image. Thus,

[TABLE]

and its commutative image is again $\check{\Phi}_{u}(X)=F_{{\rm ev}(\bar{u})}(X)=F_{{\rm ev}(u)}(X)$.

For a permutation $\sigma$ and a Dyck graph $G$, define

[TABLE]

where $\bar{G}$ is the mirror image of $G$ (which amounts to relabeling the vertices by $i\mapsto n+1-i$).

Here are the non-trivial examples of $\min^{\prime}_{G}$ for $n=3$:

[TABLE]

With these definitions, Proposition 8.3 translates into:

Proposition 8.5**.**

The noncommutative $t$-chromatic polynomial is $\check{\Phi}$-positive:

[TABLE]

For example,

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Theorem 8.6**.**

The noncommutative unicellular LLT polynomials are $\check{\Phi}$-positive:

[TABLE]

where $G_{\emptyset}$ is the graph with ($n$ vertices, $n$ omitted) and no edges.

*Proof – *If ${\bf M}_{v}$ occurs in $\check{\Phi}_{u}$, then ${\rm asc}_{G}(v)={\rm asc}_{G}(u)$ for any graph $G$. Also, since $\min^{\prime}_{G_{\emptyset}}(\sigma)$ is $\bar{v}$ where $v$ is the minimal element of ${\rm DST\,}(\bar{\sigma})$,

[TABLE]

so that

[TABLE]

For example,

[TABLE]

[TABLE]

[TABLE]

[TABLE]