Séminaire Lotharingien de Combinatoire, 93B.80 (2025), 12 pp.
Ian Cavey and Yi-Lin Lee
Domino Tilings and Macdonald Polynomials
Abstract.
There is a general bijection between domino tilings of planar regions in the square lattice and families of non-intersecting Schr\"oder-like paths contained in the region. Motivated by this bijection, we study domino tilings of certain regions Rλ, indexed by partitions λ, 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 λ = (nn) is a square shape, domino tilings of Rλ 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.
Received: November 15, 2024.
Accepted: February 15, 2025.
Final version: April 1, 2025.
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