LLT polynomials, chromatic quasisymmetric functions and graphs with cycles

Abstract

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 has not been studied before. A lot of the combinatorics regarding unit interval graphs carries over to this more general setting, and we prove several statements regarding the $e$-coefficients of chromatic quasisymmetric functions and LLT polynomials. In particular, we believe that certain $e$-positivity conjectures hold in all these families above. Furthermore, we study vertical-strip LLT polynomials, for which there is no natural chromatic quasisymmetric counterpart. These polynomials are essentially modified Hall--Littlewood polynomials, and are therefore of special interest. In this more general framework, we are able to give a natural combinatorial interpretation for the $e$-coefficients for the line graph and the cycle graph, in both the chromatic and the LLT setting.

39 pages

References in corpus (3)

• A second proof of the Shareshian--Wachs conjecture, by way of a new Hopf algebra
• The cohomology of abelian Hessenberg varieties and the Stanley-Stembridge conjecture
• A directed graph generalization of chromatic quasisymmetric functions

Cited by in corpus (16)

• A combinatorial expansion of vertical-strip LLT polynomials in the basis of elementary symmetric functions
• LLT polynomials, elementary symmetric functions and melting lollipops
• A directed graph generalization of chromatic quasisymmetric functions
• Cyclic sieving, skew Macdonald polynomials and Schur positivity
• $P$-partitions and $p$-positivity
• A symmetric function of increasing forests
• Chromatic symmetric functions of Dyck paths and q-rook theory
• The Newton polytope and Lorentzian property of chromatic symmetric functions
• (GL_k x S_n)-modules and nabla of hook-indexed Schur functions
• The cyclic sieving phenomenon on circular Dyck paths
• Down-up algebras and chromatic symmetric functions
• An area-depth symmetric $q,t$-Catalan polynomial
• Chromatic symmetric functions of Dyck paths and q-rook theory (extended abstract)
• LLT cumulants and graph coloring
• A unipotent realization of the chromatic quasisymmetric function
• Splines on Cayley Graphs of the Symmetric Group