In his study of the four colour problem, Birkhoff showed that the number of ways to colour a graph with k colours is a polynomial chi(k), which he called the chromatic polynomial. Later, Stanley defined the chromatic symmetric function X(x_1, x_2, ... ), which is a multivariable lift of the chromatic polynomial so that when the first k variables are set to 1, it recovers chi(k). This can be further lifted to a homological setting; we can construct a chain complex of graded S_n-modules whose homology has a bigraded Frobenius characteristic that recovers X upon setting q=t=1.

In this talk, we will explain the construction of the homology, discuss some new results regarding the strength of the homology as a graph invariant, and state some surprising conjectures regarding integral symmetric homology for graphs. This is based on joint work with Chandler, Sazdanovic, and Stella.

Given a graph and a set of colors, a coloring 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 Recently, Abreu-Nigro generalized the former connection for the Shareshian-Wachs \(q\)-analogue, and in unpublished work, Guay-Paquet generalized the latter.

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. Along the way, we will also discuss \(q\)-hit numbers, two variants of their statistic, and some deletion-contraction relations. This is recent work with Alejandro H. Morales and Greta Panova.

A Kronecker coefficient is a non-negative integer that depends on three partitions of a natural number n. It is the multiplicity of an irreducible representation of the symmetric group of degree n in the tensor (or Kronecker) product of two other irreducible representations of the same group.

The study of ways of computing Kronecker coefficients is an important topic on algebraic combinatorics. It was initiated by Francis Murnaghan more than eighty years ago. Several tools have been used to try to understand them, notably from representation theory, symmetric functions theory and Borel-Weil theory. These numbers generalize the well-known Littlewood-Richardson coefficients, but are still very far to be fully grasped.

It is known that each Kronecker coefficient can be described as an alternating sum of numbers of integer points in convex polytopes.

In this talk we present a new family of polytopes that permit very fast computations on Kronecker coefficients associated to partitions with few parts. This family provides, in particular, insight into some properties of Kronecker coefficients as well as into Murnaghan stability.

The Castelnuovo-Mumford regularity of a graded module provides a measure of how complicated its minimal free resolution is. In work with Rajchogt, Ren, Robichaux, and St. Dizier, we noted that the CM-regularity of matrix Schubert varieties can be easily obtained by knowing the degree of the corresponding Grothendieck polynomial. Furthermore, we gave explicit, combinatorial formulas for these degrees for symmetric Grothendieck polynomials. In this talk, I will present a general degree formula for Grothendieck polynomials. This is joint work with Oliver Pechenik and David Speyer.

Many combinatorial objects, such as matroids, graphs, and posets, can be realized as generalized permutahedra - a beautiful family of polytopes. This realization respects the natural multiplication of these objects as well as natural "breaking" operations. Surprisingly many of the important invariants of these objects, when viewed as functions on polytopes, satisfy an inclusion-exclusion formula with respect to subdivisions. Functions that satisfy this formula are known as valuations. In this talk, I will discuss recent work with Federico Ardila that completely describes the relationship between the algebraic structure on generalized permutahedra and valuations. Our main contribution is a new easy-to-apply method that converts simple valuations into more complicated ones. We also describe an universality property of the Hopf monoid of indicator functions of generalized permutahedra that extends the relationship between Combinatorial Hopf algebras and QSYM.

Combinatorial species provide a unified framework to study families of combinatorial objects. If the family of combinatorial objects has natural operations to merge and break structures, the corresponding species becomes a Hopf monoid. In this talk, we present a novel definition of type B Hopf monoids. In the same spirit as the work of Bergeron and Choquette, we consider structures over finite sets with a fixed-point free involution. However, instead of defining a monoidal structure on the category of type B species, our construction involves an action of the monoidal category of (standard) species on the category of Type B species. We present the basic definitions and some examples of Type B Hopf monoids constructed from (type B) set compositions, (symplectic) matroids, and (type B) generalized permutahedra. This is work in progress with M. Aguiar.

Classical Howe duality provides a representation theoretic framework for classical invariant theory. In the classical Howe duality the general linear group \(GL_n\) is dual to \(GL_k\) when acting on the polynomial ring on variables \(x_{ij}\) where \(1\leq i\leq n\) and \(1\leq j \leq k\). In this talk we restrict the action of \(GL_n\) to the group of permutation matrices and show that the Howe dual is an algebra whose basis is indexed by multiset partition algebra.

The peak algebra is originally introduced by Stembridge using enriched \(P\)-partitions. Using the character theory by Aguiar-Bergeron-Sottile, the peak algebra is also the image of \(\Theta\), the universal morphism between certain combinatorial Hopf algebras. We introduce a shuffle basis of quasi-symmetric functions that has a shuffle-like Hopf structure and is also the eigenfunctions of \(\Theta\). Using this new basis, we extend the notion of peak algebras and theta maps to shuffle, tensor and symmetric algebras. As examples, we study the peak algebras of symmetric functions in non-commuting variables and the graded associated Hopf algebra on permutations.