Algebraic Combinatorics Seminar - Laura Colmenarejo

Title: Chromatic symmetric functions of Dyck paths and $q$-rook theory

Abstract:

Given a graph and a set of colors, a coloring of the graph is a function that associates each vertex in the graph with a color. In 1995, Stanley generalized this definition to symmetric functions by looking at the number of times each color is used and extending the set of colors to $\mathbb{Z}^+$. In 2012, Shareshian and Wachs introduced a refinement of the chromatic functions for ordered graphs as $q$-analogues.

In the particular case of Dyck paths, Stanley and Stembridge described the connection between chromatic symmetric functions of abelian Dyck paths and square hit numbers, and Guay-Paquet described their relation to rectangular hit numbers.

Recently, Abreu-Nigru generalized the former connection for the Shareshian-Wachs $q$-analogue, and in unpublished work, Guay-Paquet generalized the latter. Both of these generalizations use the Garsia-Remmel $q$-hit numbers.  

In this talk, I want to give an overview of the framework and present another proof of Guay-Paquet's identity using $q$-rook theory and use it to give a new proof of the Abreu-Nigru identity. This is recent work with Alejandro H. Morales and Greta Panova.