A Shuffling Theorem for Centrally Symmetric Tilings

Abstract

Rohatgi and the author recently proved a shuffling theorem for lozenge tilings of `doubly-dented hexagons' (arXiv:1905.08311). The theorem can be considered as a hybrid between two classical theorems in the enumeration of tilings: MacMahon's theorem about centrally symmetric hexagons and Cohn-Larsen-Prop's theorem about semihexagons with dents. In this paper, we consider a similar shuffling theorem for the centrally symmetric tilings of the doubly-dented hexagons. Our theorem also implies a conjecture posed by the author in arXiv:1803.02792 about the enumeration of centrally symmetric tilings of hexagons with three arrays of triangular holes. This enumeration, in turn, can be considered as a common generalization of (a tiling-equivalent version of) Stanley's enumeration of self-complementary plane partitions and Ciucu's work on symmetries of the shamrock structure. Moreover, our enumeration also confirms a recent conjecture posed by Ciucu in arXiv:1906.02951.

arXiv commentSecond version: 26 pages and many pictures

References in corpus (7)

• The shape of a typical boxed plane partition
• A shuffling theorem for lozenge tilings of doubly-dented hexagons
• Tiling enumeration of doubly-intruded halved hexagons
• A Shuffling Theorem for Reflectively Symmetric Tilings
• Lozenge tilings of hexagons with central holes and dents
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• Centrally symmetric tilings of fern-cored hexagons

Cited by in corpus (2)

• Ratio of tiling generating functions of semi-hexagons and quartered hexagons with dents
• Problems in the Enumeration of Tilings