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arXiv · 1905.08311

A shuffling theorem for lozenge tilings of doubly-dented hexagons

Tri Lai·Ranjan Rohatgi
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Abstract

MacMahon's theorem on plane partitions yields a simple product formula for tiling number of a hexagon, and Cohn, Larsen and Propp's theorem provides an explicit enumeration for tilings of a dented semihexagon via semi-strict Gelfand--Tsetlin patterns. In this paper, we prove a natural hybrid of the two theorems for hexagons with an arbitrary set of unit triangles removed along a horizontal axis. In particular, we show that the `shuffling' of removed unit triangles only changes the tiling number of the region by a simple multiplicative factor. Our main result generalizes a number of known enumerations and asymptotic enumerations of tilings. We also reveal connections of the main result to the study of symmetric functions and $q$-series.

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Tri Lai, Ranjan Rohatgi. 2019-05-20. A shuffling theorem for lozenge tilings of doubly-dented hexagons. https://arxiv.org/abs/1905.08311

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