The Stanley chromatic symmetric function X_G of a graph G is a symmetric function generalization of the chromatic polynomial, and has interesting combinatorial properties. We apply the ideas of Khovanov homology to construct a homology of graded S_n-modules, whose graded Frobenius series Frob_G(q,t) reduces to the chromatic symmetric function at q=t=1. This homology can be thought of as a categorification of the chromatic symmetric function, and provides a homological analogue of several familiar properties of X_G. In particular, the decomposition formula for X_G discovered recently by Orellana and Scott, and Guay-Paquet, is lifted to a long exact sequence in homology. (joint with R. Sazdanovic)

Macdonald polynomials are orthogonal polynomials associated to root systems, and in the type A case, the symmetric Macdonald polynomials are a common generalization of Schur functions, Hall-Littlewood polynomials, and Jack polynomials. We use the combinatorics of alcove walks to obtain formulas for computing products of monomials and intertwining operators of the double affine Hecke algebra, and from this, we obtain a product formula for Macdonald polynomials of general type.

In this paper, we use the combinatorics of alcove walks to give a uniform combinatorial formula for Macdonald polynomials for all Lie types. These formulas are generalizations of the formulas of Haglund-Haiman-Loehr for Macdonald polynomials of type GL(n). At q=0, these formulas specialize to the formula of Schwer for the Macdonald spherical function in terms of positive folded alcove walks, and at q=t=0, these formulas specialize to the formula for the Weyl character in terms of the Littelmann path model (in the positively folded gallery form of Gaussent-Littelmann). (joint with A. Ram)

The set of n by n upper-triangular nilpotent matrices with entries in a finite field F_q has Jordan canonical forms indexed by partitions lambda of n. We present a combinatorial formula for computing the number F_\lambda(q) of matrices of Jordan type lambda as a weighted sum over standard Young tableaux. We construct a bijection between paths in a modified version of Young's lattice and non-attacking rook placements, which leads to a refinement of the formula for F_lambda(q).

(joint with D.M. Jackson)

(joint with D. Panario, L.B. Richmond)