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arXiv · 1709.02616

A Curious Family of Binomial Determinants That Count Rhombus Tilings of a Holey Hexagon

Christoph Koutschan·Thotsaporn Thanatipanonda
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Abstract

We evaluate a curious determinant, first mentioned by George Andrews in 1980 in the context of descending plane partitions. Our strategy is to combine the famous Desnanot-Jacobi-Dodgson identity with automated proof techniques. More precisely, we follow the holonomic ansatz that was proposed by Doron Zeilberger in 2007. We derive a compact and nice formula for Andrews's determinant, and use it to solve a challenge problem that we posed in a previous paper. By noting that Andrews's determinant is a special case of a two-parameter family of determinants, we find closed forms for several one-parameter subfamilies. The interest in these determinants arises because they count cyclically symmetric rhombus tilings of a hexagon with several triangular holes inside.

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Christoph Koutschan, Thotsaporn Thanatipanonda. 2017-09-08. A Curious Family of Binomial Determinants That Count Rhombus Tilings of a Holey Hexagon. https://doi.org/10.1016/j.jcta.2019.03.001

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