A modular relation for the chromatic symmetric functions of (3+1)-free posets

Abstract

We consider a linear relation which expresses Stanley's chromatic symmetric function for a poset in terms of the chromatic symmetric functions of some closely related posets, which we call the modular law. By applying this in the context of (3+1)-free posets, we are able to reduce Stanley and Stembridge's conjecture that the chromatic symmetric functions of all (3+1)-free posets are e-positive to the case of (3+1)-and-(2+2)-free posets, also known as unit interval orders. In fact, our reduction can be pushed further to a much smaller class of posets, for which we have no satisfying characterization. We also obtain a new proof of the fact that all 3-free posets have e-positive chromatic symmetric functions.

10 pages, 1 figure

Cited by in corpus (22)

• LLT polynomials, chromatic quasisymmetric functions and graphs with cycles
• The cohomology of abelian Hessenberg varieties and the Stanley-Stembridge conjecture
• A combinatorial expansion of vertical-strip LLT polynomials in the basis of elementary symmetric functions
• A directed graph generalization of chromatic quasisymmetric functions
• Chromatic symmetric functions via the group algebra of $S_n$
• On $e$-positivity and $e$-unimodality of chromatic quasisymmetric functions
• Chromatic symmetric functions of Dyck paths and q-rook theory
• From Poset Topology to $q$-Eulerian Polynomials to Stanley's Chromatic Symmetric Functions
• Chromatic bases for symmetric functions
• A survey of recent developments on Hessenberg varieties
• Chromatic classical symmetric functions
• H-chromatic symmetric functions
• Bases of the equivariant cohomologies of regular semisimple Hessenberg varieties
• Hook-Shape Immanant Characters from Stanley-Stembridge Characters
• LLT cumulants of unicellular Young diagrams, parking functions and Schur positivity
• The $e$-positivity of two families of $(claw,2K_2)$-free graphs
• Torus actions, localization and induced representations on cohomology
• Splines on Cayley Graphs of the Symmetric Group
• A new approach to $e$-positivity for Stanley's chromatic functions
• Chromatic Signed-Symmetric Functions of Signed Graphs
• Chromatic symmetric functions of Dyck paths and q-rook theory (extended abstract)
• S-positivity of interval hypergraphs