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

Christoph Koutschan and Thotsaporn "Aek" Thanatipanonda

### 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.### Paper

The material on this webpage accompanies the article A Curious Family of Binomial Determinants That Count Rhombus Tilings of a Holey Hexagon by Christoph Koutschan and Thotsaporn "Aek" Thanatipanonda (Journal of Combinatorial Theory, Series A, vol. 166, pp. 352–381, 2019).### Supplementary electronic material

We provide a Mathematica notebook- CuriousDeterminants.nb (Mathematica notebook, last update 01.07.2018)
- CuriousDeterminants.cdf (Mathematica computable document)
- CuriousDeterminants.pdf (pdf printout of the notebook)

Additionally, we provide some precomputed results, namely the
recursive descriptions of the auxiliary functions c_{n,j} that
appear in the proofs, and the corresponding creative telescoping
relations (although their computation doesn't require an excessive
amount of time). Just unpack the zip file into the same directory where
the notebook is stored.

- precomputed.zip (1.4 MB)

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