In 1993, Stanley and Stembridge conjectured that a chromatic symmetric function of any $(3+1)$-free poset is $e$-positive. Guay-Paquet reduced the conjecture to $(3+1)$- and $(2+2)$-free posets which are also called natural unit interval orders. Shareshian and Wachs defined chromatic quasisymmetric functions, generalizing chromatic symmetric functions, and conjectured that a chromatic quasisymmetric function of any natural unit interval order is $e$-positive and $e$-unimodal.
For a given natural interval order, there is a corresponding partition $λ$ and we denote the chromatic quasisymmetric function by $X_λ$. The first author introduced local linear relations for chromatic quasisymmetric functions. In this paper, we prove a powerful generalization of the above-mentioned local linear relations, called a rectangular lemma, which also generalizes the result of Huh,Nam and Yoo. Such a lemma can be applied to describe explicit formulas for $e$-positivity of a chromatic symmetric function $X_λ$ where $λ$ is contained in a rectangle. We also suggest some conjectural formulas for $e$-positivity when $λ$ is not contained in a rectangle by applying the rectangular lemma.
The $e$-positivity conjecture and the $e$-unimodality conjecture of chromatic quasisymmetric functions are proved for some classes of natural unit interval orders. Recently, J. Shareshian and M. Wachs introduced chromatic quasisymmetric functions as a refinement of Stanley's chromatic symmetric functions and conjectured the $e$-positivity and the $e$-unimodality of these functions. The $e$-positivity of chromatic quasisymmetric functions implies the $e$-positivity of corresponding chromatic symmetric functions, and our work resolves Stanley's conjecture on chromatic symmetric functions of $(3+…
The chromatic symmetric function $X_G$ of a graph $G$ was introduced by Stanley. In this paper we introduce a quasisymmetric generalization $X^k_G$ called the $k$-chromatic quasisymmetric function of $G$ and show that it is positive in the fundamental basis for the quasisymmetric functions. Following the specialization of $X_G$ to $χ_G(λ)$, the chromatic polynomial, we also define a generalization $χ^k_G(λ)$ and show that evaluations of this polynomial for negative values generalize a theorem of Stanley relating acyclic orientations to the chromatic polynomial.
We express the integral form Macdonald polynomials as a weighted sum of Shareshian and Wachs' chromatic quasisymmetric functions of certain graphs. Then we use known expansions of these chromatic quasisymmetric functions into Schur and power sum symmetric functions to provide Schur and power sum formulas for the integral form Macdonald polynomials. Since the (integral form) Jack polynomials are a specialization of integral form Macdonald polynomials, we obtain analogous formulas for Jack polynomials as corollaries.
Stanley defined the chromatic symmetric function of a graph, and Shareshian and Wachs introduced a refinement, namely the chromatic quasisymmetric function of a labeled graph. In this paper, we define the chromatic quasisymmetric function of a directed graph, which agrees with the Shareshian-Wachs definition in the acyclic case. We give an F-basis expansion for all digraphs in terms of a permutation statistic, which we call G-descents. We use this expansion to derive a p-positivity formula for all digraphs with symmetric chromatic quasisymmetric functions. We show that the chromatic quasisymme…
We introduce a quasisymmetric class function associated with a group acting on a double poset or on a directed graph. The latter is a generalization of the chromatic quasisymmetric function of a digraph introduced by Ellzey, while the latter is a generalization of a quasisymmetric function introduced by Grinberg. We prove representation-theoretic analogues of classical and recent results, including $F$-positivity, and combinatorial reciprocity theorems. We also deduce results for orbital quasisymmetric functions. We also study a generalization of the notion of strongly flawless sequences.
The chromatic quasisymmetric function of a graph was introduced by Shareshian and Wachs as a refinement of Stanley's chromatic symmetric function. An explicit combinatorial formula, conjectured by Shareshian and Wachs, expressing the chromatic quasisymmetric function of the incomparability graph of a natural unit interval order in terms of power sum symmetric functions, is proven. The proof uses a formula of Roichman for the irreducible characters of the symmetric group.
Chromatic quasisymmetric functions of labeled graphs were defined by Shareshian and Wachs as a refinement of Stanley's chromatic symmetric functions. In this extended abstract, we consider an extension of their definition from labeled graphs to directed graphs, suggested by Richard Stanley. We obtain an F-basis expansion of the chromatic quasisymmetric functions of all digraphs and a p-basis expansion for all symmetric chromatic quasisymmetric functions of digraphs, extending work of Shareshian-Wachs and Athanasiadis. We show that the chromatic quasisymmetric functions of proper circular arc d…
We introduce a quasisymmetric refinement of Stanley's chromatic symmetric function. We derive refinements of both Gasharov's Schur-basis expansion of the chromatic symmetric function and Chow's expansion in Gessel's basis of fundamental quasisymmetric functions. We present a conjectural refinement of Stanley's power sum basis expansion, which we prove in special cases. We describe connections between the chromatic quasisymmetric function and both the $q$-Eulerian polynomials introduced in our earlier work and, conjecturally, representations of symmetric groups on cohomology of regular semisimp…
We study the chromatic quasisymmetric class function of a linearized combinatorial Hopf monoid. Given a linearized combinatorial Hopf monoid $H$, and an $H$-structure $h$ on a set $N$, there are proper colorings of $h$, generalizing graph colorings and poset partitions. We show that the automorphism group of $h$ acts on the set of proper colorings. The chromatic quasisymmetric class function enumerates the fixed points of this action, weighting each coloring with a monomial. For the Hopf monoid of graphs this invariant generalizes Stanley's chromatic symmetric function and specializes to the o…
In this work, we generalize and utilize the linear relations of LLT polynomials introduced by Lee \cite{Lee}. By using the fact that the chromatic quasisymmetric functions and the unicellular LLT polynomials are related via plethystic substitution and thus they satisfy the same linear relations, we can apply the linear relations to both sets of functions. As a result, in the chromatic quasisymmetric function side, we find a class of $e$-positive graphs, called \emph{melting lollipop graphs}, and explicitly prove the $e$-unimodality. In the unicellular LLT side, we obtain Schur expansion formul…
We use a Dyck path model for unit-interval graphs to study the chromatic quasisymmetric functions introduced by Shareshian and Wachs, as well as vertical strip --- in particular, unicellular LLT polynomials. We show that there are parallel phenomena regarding $e$-positivity of these two families of polynomials. In particular, we give several examples where the LLT polynomials behave like a "mirror image" of the chromatic quasisymmetric counterpart. The Dyck path model is also extended to circular arc digraphs to obtain larger families of polynomials. This circular extensions of LLT polynomials…
We discuss three distinct topics of independent interest; one in enumerative combinatorics, one in symmetric function theory, and one in algebraic geometry. The topic in enumerative combinatorics concerns a q-analog of a generalization of the Eulerian polynomials, the one in symmetric function theory deals with a refinement of the chromatic symmetric functions of Stanley, and the one in algebraic geometry deals with Tymoczko's representation of the symmetric group on the cohomology of the regular semisimple Hessenberg variety of type A. Our purpose is to explore some remarkable connections bet…
We study the chromatic symmetric function on graphs, and show that its kernel is spanned by the modular relations. We generalize this result to the chromatic quasisymmetric function on hypergraphic polytopes, a family of generalized permutahedra. We use this description of the kernel of the chromatic symmetric function to find other graph invariants that may help us tackle the tree conjecture.
Given an element in a finite-dimensional real vector space, $V$, that is a nonnegative linear combination of basis vectors for some basis $B$, we compute the probability that it is furthermore a nonnegative linear combination of basis vectors for a second basis, $A$. We then apply this general result to combinatorially compute the probability that a symmetric function is Schur-positive (recovering the recent result of Bergeron--Patrias--Reiner), $e$-positive or $h$-positive. Similarly we compute the probability that a quasisymmetric function is quasisymmetric Schur-positive or fundamental-posi…
In this article we show how to compute the chromatic quasisymmetric function of indifference graphs from the modular law introduced by Guay-Paquet. We provide an algorithm which works for any function that satisfies this law, such as unicellular LLT polynomials. When the indifference graph has bipartite complement it reduces to a planar network, in this case, we prove that the coefficients of the chromatic quasisymmetric function in the elementary basis are positive unimodal polynomials and characterize them as certain $q$-hit numbers (up to a factor). Finally, we discuss the logarithmic conca…
Motivated by the study of Macdonald polynomials, J. Haglund and A. Wilson introduced a nonsymmetric polynomial analogue of the chromatic quasisymmetric function called the \emph{chromatic nonsymmetric polynomial} of a Dyck graph. We give a positive expansion for this polynomial in the basis of fundamental slide polynomials using recent work of Assaf-Bergeron on flagged $(P,ρ)$-partitions. We then derive the known expansion for the chromatic quasisymmetric function of Dyck graphs in terms of Gessel's fundamental basis by taking a backstable limit of our expansion.
We consider families of quasisymmetric functions with the property that if a symmetric function $f$ is a positive sum of functions in one of these families, then f is necessarily a positive sum of Schur functions. Furthermore, in each of the families studied, we give a combinatorial description of the Schur coefficients of $f$. We organize six such families into a poset, where functions in higher families in the poset are always positive integer sums of functions in each of the lower families. This poset includes the Schur functions, the quasisymmetric Schur functions, the fundamental quasisym…
We introduce a weighted quasisymmetric enumerator function associated to generalized permutohedra. It refines the Billera, Jia and Reiner quasisymmetric function which also includes the Stanley chromatic symmetric function. Beside that it carries information of face numbers of generalized permutohedra. We consider more systematically the cases of nestohedra and matroid base polytopes.
We investigate the $e$-positivity and Schur positivity of the chromatic symmetric functions of some spider graphs with three legs. We obtain the positivity classification of all broom graphs and that of most double broom graphs. The methods involve extracting particular $e$-coefficients of the chromatic symmetric function of these graphs with the aid of Orellana and Scott's triple-deletion property, and using the combinatorial formula of Schur coefficients by examining certain special rim hook tabloids. We also propose some conjectures on the $e$-positivity and Schur positivity of trees.
We show that no tree on twenty vertices with maximum degree ten has Schur positive chromatic symmetric function, thereby providing a counterexample to a conjecture from the paper "Schur and e-positivity of trees and cut vertices".
For a hypergraphic polytope there is a weighted quasisymmetric function which enumerates positive integer points in its normal fan and determines its f-polynomial. This quasisymmetric function invariant of hypergraphs extends the Stanley chromatic symmetric function of simple graphs. We consider a certain combinatorial Hopf algebra of hypergraphs and show that universal morphism to quasisymmetric functions coincides with this enumerator function.
We investigate chromatic symmetric functions in the relation to the algebra $Γ$ of symmetric functions generated by Schur $Q$-functions. We construct natural bases of $Γ$ in terms of chromatic symmetric functions. We also consider the $p$-positivity of skew Schur $Q$-functions and find a class of $p$-positive ribbon Schur $Q$-functions, making a conjecture that they are \emph{all}. We include many concrete computational results that support our conjecture.
In 1995 Stanley introduced the chromatic symmetric function $X_G$ of a graph $G$, whose $e$-positivity and Schur-positivity has been of large interest. In this paper we study the relative $e$-positivity and Schur-positivity between connected graphs on $n$ vertices. We define and investigate two families of posets on distinct chromatic symmetric functions. The relations depend on the $e$-positivity or Schur-positivity of a weighed subtraction between $X_G$ and $X_H$. We find a biconditional condition between $e$-positivity or Schur-positivity and the relation to the complete graph. This gives a…
We give a proof of Stanley-Stembridge conjecture on chromatic symmetric functions for the class of all unit interval graphs with independence number 3. That is, we show that the chromatic symmetric function of the incomparability graph of a unit interval order in which the length of a chain is at most 3 is positively expanded as a linear sum of elementary symmetric functions.
Universite Claude Bernard Lyon 1, CNRS, Ecole Centrale de Lyon, INSA Lyon, Université Jean Monnet, ICJ UMR5208（里昂第一大学、法国国家科学研究中心、里昂中央理工学院、里昂国立应用科学学院、让·莫内大学、ICJ联合研究实验室）；Department of Mathematics, University of Toronto（多伦多大学数学系）；Department of Mathematical and Computational Sciences, University of Toronto Mississauga（多伦多大学密西沙加分校数学与计算科学系）
Richard P. Stanley conjectured that finite trees can be distinguished by their chromatic symmetric functions. In this paper, we prove an analogous statement for posets: Finite rooted trees can be distinguished by their order quasisymmetric functions.
In this paper, we study positivity phenomena for the $e$-coefficients of Stanley's chromatic function of a graph. We introduce a new combinatorial object: the {\em correct} sequences of unit interval orders, and using these, in certain cases, we succeed to construct combinatorial models of the coefficients appearing in Stanley's conjecture. Our main result is the proof of positivity of the coefficients $c_{n-k,1^k}$, $c_{n-2,2}$, $c_{n-3,2,1}$ and $c_{2^k,1^{n-2k}}$ of the expansion of the chromatic symmetric function in terms of the basis of the elementary symmetric polynomials for the case o…
We provide a formula for every Schur coefficient in the chromatic symmetric function of a graph in terms of special rim hook tabloids. This formula is useful in confirming the non-Schur positivity of the chromatic symmetric function of a graph, especially when Stanley's stable partition method does not work. As applications, we determine Schur positive fan graphs and Schur positive complete tripartite graphs. We show that any squid graph obtained by adding $n$ leaves to a common vertex on an $m$-vertex cycle is not Schur positive if $m\ne 2n-1$, and conjecture that neither are the squid graphs…
Richard P. Stanley defined the chromatic symmetric function of a simple graph and has conjectured that every tree is determined by its chromatic symmetric function. Recently, Takahiro Hasebe and the author proved that the order quasisymmetric functions, which are analogs of the chromatic symmetric functions, distinguish rooted trees. In this paper, using a similar method, we prove that the chromatic symmetric functions distinguish trivially perfect graphs. Moreover, we also prove that claw-free cographs, that is, $ \{K_{1,3},P_{4}\} $-free graphs belong to a known class of $ e $-positive graph…
Determining whether two graphs are isomorphic is an important and difficult problem in graph theory. One way to make progress towards this problem is by finding and studying graph invariants that distinguish large classes of graphs. Stanley conjectured that his chromatic symmetric function distinguishes all trees, which has remained unresolved. Recently, Hasebe and Tsujie introduced an analogue of Stanley's function for posets, called the strict order quasisymmetric function, and proved that it distinguishes all rooted trees. In this paper, we devise a procedure to explicitly reconstruct a roo…
The chromatic symmetric function $X_H$ of a hypergraph $H$ is the generating function for all colorings of $H$ so that no edge is monochromatic. When $H$ is an ordinary graph, it is known that $X_H$ is positive in the fundamental quasisymmetric functions $F_S$, but this is not the case for general hypergraphs. We exhibit a class of hypergraphs $H$ --- hypertrees with prime-sized edges --- for which $X_H$ is $F$-positive, and give an explicit combinatorial interpretation for the $F$-coefficients of $X_H$.
Over the past years, major attention has been drawn to the question of identifying Schur-positive sets, i.e. sets of permutations whose associated quasisymmetric function is symmetric and can be written as a non-negative sum of Schur symmetric functions. The set of arc permutations, i.e. the set of permutations $π$ in $S_n$ such that for any $1\leq j \leq n$, $\{π(1),π(2),\dots,π(j)\}$ is an interval in $\mathbb{Z}_n$ is one of the most noticeable examples. This paper introduces a new type B extension of Schur-positivity to signed permutations based on Chow's quasisymmetric functions and gener…
We make a systematic study of a new combinatorial construction called a dual equivalence graph. We axiomatize these graphs and prove that their generating functions are symmetric and Schur positive. This provides a universal method for establishing the symmetry and Schur positivity of quasisymmetric functions.
This article is dedicated to the study of positivity phenomena for the chromatic symmetric function of a graph with respect to various bases of symmetric functions. We give a new proof of Gasharov's theorem on the Schur-positivity of the chromatic symmetric function of a $(3 + 1)$-free poset. We present a combinatorial interpretation of the Schur-coefficients in terms of planar networks. Compared to Gasharov's proof, it gives a clearer visual illustration of the cancellation procedures and is quite similar in spirit to the proof of monomial positivity of Schur functions via the Lindstrom-Gesse…
We prove the s-positivity of the chromatic functions of interval hypergraphs and a conjecture of Taylor~\cite{Taylor15} as a consequence. We also give a new proof of Gasharov's theorem on the s-positivity of the chromatic symmetric functions of $(3+1)$-free posets.
In Stanley's seminal 1995 paper on the chromatic symmetric function, he stated that there was no known graph that was not contractible to the claw and whose chromatic symmetric function was not $e$-positive, namely, not a positive linear combination of elementary symmetric functions. We resolve this by giving infinite families of graphs that are not contractible to the claw and whose chromatic symmetric functions are not $e$-positive. Moreover, one such family is additionally claw-free, thus establishing that the $e$-positivity of chromatic symmetric functions is in general not dependent on th…
Recently, Hyatt introduced some colored Eulerian quasisymmetric function to study the joint distribution of excedance number and major index on colored permutation groups. We show how Hyatt's generating function formula for the fixed point colored Eulerian quasisymmetric functions can be deduced from the Decrease value theorem of Foata and Han. Using this generating function formula, we prove two symmetric function generalizations of the Chung-Graham-Knuth symmetrical Eulerian identity for some flag Eulerian quasisymmetric functions, which are specialized to the flag excedance numbers. Combina…
We introduce the notion of a scheduling problem which is a boolean function $S$ over atomic formulas of the form $x_i \leq x_j$. Considering the $x_i$ as jobs to be performed, an integer assignment satisfying $S$ schedules the jobs subject to the constraints of the atomic formulas. The scheduling counting function counts the number of solutions to $S$. We prove that this counting function is a polynomial in the number of time slots allowed. Scheduling polynomials include the chromatic polynomial of a graph, the zeta polynomial of a lattice, the Billera-Jia-Reiner polynomial of a matroid. To an…
We introduce a family of quasisymmetric functions called {\em Eulerian quasisymmetric functions}, which have the property of specializing to enumerators for the joint distribution of the permutation statistics, major index and excedance number on permutations of fixed cycle type. This family is analogous to a family of quasisymmetric functions that Gessel and Reutenauer used to study the joint distribution of major index and descent number on permutations of fixed cycle type. Our central result is a formula for the generating function for the Eulerian quasisymmetric functions, which specialize…
Let $P$ be a poset, $inc(P)$ its incomparability graph, and $X_{inc(P)}$ the corresponding chromatic symmetric function, as defined by Stanley in {\em Adv. Math.}, {\bf 111} (1995) pp.~166--194. Certain conditions on $P$ imply that the expansions of $X_{inc(P)}$ in standard symmetric function bases yield coefficients which have simple combinatorial interpretations. By expressing these coefficients as character evaluations, we extend several of these interpretations to {\em all} posets $P$. Consequences include new combinatorial interpretations of the permanent and other immanants of totally no…
Two celebrated conjectures in chromatic symmetric function theory concern the positivity chromatics symmetric functions of claw-free graphs. Here we extend the claw-free idea to general graphs and consider the e-positivity question for H-free graphs where H = {claw, F} and H={claw, F, co-F}, where F is a four-vertex graph. We settle the question for all cases except H={claw, co-diamond}, and we provide some partial results in that case.
