On $e$-Positivity and $e$-Unimodality of Chromatic Quasi-symmetric Functions

Abstract

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 p3`1q-free posets for two classes of natural unit interval orders.2010 Mathematics Subject Classification. Primary 05E05; Secondary 05C15, 05C25 .

Schur and $e$-Positivity of Trees and Cut Vertices

“…Many partial results regarding chromatic symmetric functions have been obtained such as when the graph involved is the path or the cycle [6,22,27], when the graph is formed from complete graphs [3,11,14], or when a graph avoids another [7,10,13,25]. These proofs have not always worked directly with the chromatic symmetric function.…”

“…These proofs have not always worked directly with the chromatic symmetric function. Instead, sometimes generalizations of the chromatic symmetric function have been employed such as to quasisymmetric functions [3,14,21] and noncommutative symmetric functions [11].…”

Chromatic symmetric functions in noncommuting variables revisited

“…This is not the only expression for Y G in terms of the power sum symmetric functions in NCSym, and it is the next one that will be useful for our second result. For it, given a graph G and edge set S ⊆ E(G) we define π(S) to be the set partition whose blocks are determined by the vertex labels of the connected components of G restricted to the edges in S. For example, in Example 2.7 if S = {(1, 3), (1, 4), (7,8)} then π(S) = 134/2/5/6/78.…”

“…The elementary symmetric functions arise in a variety of contexts, and one of the best known is the fundamental theorem of symmetric functions, which roughly states that Sym is generated by the elementary symmetric functions indexed by positive integers. While the (3 + 1)-conjecture still stands, there has been much progress made towards it, for example, [7,10,11,12,13,15,16,17,18,20,30,32], and the related question of when X G is a positive linear combination of Schur functions [14,23,25,27]. Most of these results have been achieved by working directly with X G , however, there has been notable success in employing its generalization to quasisymmetric functions [7,25] and symmetric functions in noncommuting variables Y G [10,16].…”

A deletion–contraction relation for the chromatic symmetric function

“…The chromatic symmetric function X G of a graph G was introduced by Stanley in 1995 [20] as a generalization of the chromatic polynomial χ G (x). It is defined as X G (x 1 , x 2 , ...) = κ v∈V (G) x κ (v) where the sum ranges over all proper colorings κ of G. Recent research on X G has focused on (among other topics) the Stanley-Stembridge conjecture that the chromatic symmetric function of the incomparability graph of a (3 + 1)-free poset is e-positive [2,4,5,8,11,20], the related conjecture that the chromatic symmetric function of a claw-free graph is s-positive [9,17,18], and the conjecture that X G distinguishes nonisomorphic trees [1,12]. Other results have extended the definition of X G to include quasisymmetric functions [7,23] or noncommuting variables [6,10].…”