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 containing all computations that constitute the proofs of the determinant evaluations in our article, and the formulas of our other results and conjectures in Mathematica format. Detailed explanations in the notebook are given in order to help the reader understand the technical details of the computations. In addition, we recommend to read our previous article Advanced computer algebra for determinants for more explanations of the general proof strategy. For the execution of the notebook the RISC combinatorics software packages Guess and HolonomicFunctions are required. Both can be accessed at the same time by installing the RISCErgoSum bundle.

Additionally, we provide some precomputed results, namely the recursive descriptions of the auxiliary functions cn,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.

Contact: