Domino Tilings, Domino Shuffling, and the Nabla Operator , Yi-Lin Lee (Indiana University Bloomington) -- January 30th, 2025

In this talk, I will present a $q,t$-generalization of domino tilings of certain regions $R_\lambda$, indexed by partitions $\lambda$, weighted according to generalized area and dinv statistics. These statistics arise from the $q,t$-Catalan combinatorics and Macdonald polynomials. We present a formula for the generating polynomial of these domino tilings in terms of the Bergeron-Garsia nabla operator. When $\lambda = (n^n)$ is a square shape, domino tilings of $R_\lambda$ are equivalent to those of the Aztec diamond of order $n$. In this case, we give a new product formula for the resulting polynomials by domino shuffling and its connection with alternating sign matrices. In particular, we obtain a combinatorial proof of the joint symmetry of the generalized area and dinv statistics. This is based on the joint work with Ian Cavey.

The seminar co-organizers are Ian Cavey, Andrew Hardt, David Keating, and Alexander Yong.