Abstract
We study the weighted partition function for lozenge tilings, with weights given by multivariate rational functions originally defined by Morales, Pak and Panova (2019) in the context of the factorial Schur functions. We prove that this partition function is symmetric for large families of regions. We employ both combinatorial and algebraic proofs.
Highlights
Hidden symmetries are pervasive across the natural sciences, but are always a delight whenever discovered
When a hidden symmetry is discovered for a well-known combinatorial structure, it is as surprising as it is puzzling, since this points to a rich structure which yet to be understood
Our approach to a multivariate deformation of Pabc is based on the recent work [MPP3] in Algebraic Combinatorics, in turn inspired by the extensive study of the cohomology of the Grassmannian
Summary
Hidden symmetries are pervasive across the natural sciences, but are always a delight whenever discovered. Our approach to a multivariate deformation of Pabc is based on the recent work [MPP3] in Algebraic Combinatorics, in turn inspired by the extensive study of the (equivariant) cohomology of the Grassmannian. To set this up, recall that the lozenge tilings of H a, b, c are in bijection with collections of non-intersecting paths in the rectangle, see Figure 1. Recall that the lozenge tilings of H a, b, c are in bijection with collections of non-intersecting paths in the rectangle, see Figure 1 These lattice paths are in bijection with the excited diagrams, giving a connection to the Naruse hook-length formula [MPP1, MPP2] the number of standard Young tableaux of skew shapes.
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