Research in Algebraic Combinatorics

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The research supported by this grant is in algebraic combinatorics, which is an area of mathematics that seeks to develop connections between combinatorics (the science of counting, arranging, and analyzing concrete discrete configurations) and fields of pure mathematics that involve sophisticated abstract algebraic structures. The idea is to use these connections to gain deeper insights and solve problems in combinatorics and in the other fields. The discrete configurations that are studied in combinatorics arise in various fields of mathematics, computer science, physics, biology, and engineering; DNA sequences, phylogenetic trees, and communications networks are all examples of discrete configurations. Combinatorial methods are playing an increasing role in these fields.A common thread running through the various parts of this project is that of palindromicity and unimodality of polynomials. Many important enumerative sequences arising in algebra, combinatorics, and geometry are palindromic and unimodal, but proving unimodality can be quite challenging. Proofs of unimodality appearing in the literature have made striking use of combinatorial, analytic, algebraic, and algebro-geometric techniques. In the PI's current work, unimodality issues have led to the discovery of intriguing connections between certain combinatorial and geometric structures and to the discovery of elegant new q-analogs of classical enumerative formulas. For example, palindromicity and unimodality of a generalized Eulerian polynomial (due to De Mari, Procesi and Shayman) play a role in the PI's and Shareshian's ongoing work on chromatic quasisymmetric functions, which are a refinement of Stanley's chromatic symmetric functions. One aim of this project is to prove a conjecture of the PI and Shareshian on a relationship between the chromatic quasisymmetric functions and Tymoczko's representation of the symmetric group on the cohomology of the regular semisimple Hessenberg variety of type A. This could establish, among other things, a longstanding e-positivity conjecture of Stanley and Stembridge for chromatic symmetric functions. A property stronger than palindromicity and unimodality, known as gamma-positivity, is of current interest in combinatorics and discrete geometry, as many important classes of enumerative polynomials are gamma-positive. Part of this project is concerned with a q-analog of gamma-positivity recently studied in joint work with Dilks and Krattenthaler. Another part of the project continues the work of the PI on the interplay between poset topology and enumerative combinatorics. One aim is to prove a conjecture of the PI and her former student Gonzalez D'Leon connecting the topology of certain subposets of the poset of weighted partitions to gamma-positive h-polynomials of graph associahedra.