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

TL;DR

This paper evaluates a special determinant related to counting rhombus tilings of a hexagon with holes, using advanced identities and automated proof techniques to derive explicit formulas and solve related combinatorial problems.

Contribution

It provides a new compact formula for Andrews's determinant and extends it to a family of determinants counting symmetric rhombus tilings with holes.

Findings

Derived a closed-form formula for Andrews's determinant.

Solved a previously posed challenge problem using the formula.

Found explicit formulas for subfamilies of the determinant family.

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.