Binomial Determinants for Tiling Problems Yield to the Holonomic Ansatz

TL;DR

This paper derives closed-form binomial determinant formulas related to cyclically symmetric rhombus tilings with holes, extending previous work to negative Kronecker deltas, and resolves longstanding conjectures using the holonomic ansatz and computer algebra.

Contribution

It extends the study of binomial determinants with Kronecker deltas to negative cases and applies the holonomic ansatz for symbolic summation, solving open conjectures.

Findings

Closed-form expressions for binomial determinants with signed Kronecker deltas.

New determinantal relationships uncovered through symbolic summation.

Resolution of all remaining open conjectures related to these determinants.

Abstract

We present and prove closed form expressions for some families of binomial determinants with signed Kronecker deltas that are located along an arbitrary diagonal in the corresponding matrix. They count cyclically symmetric rhombus tilings of hexagonal regions with triangular holes. We extend a previous systematic study of these families, where the locations of the Kronecker deltas depended on an additional parameter, to families with negative Kronecker deltas. By adapting Zeilberger's holonomic ansatz to make it work for our problems, we can take full advantage of computer algebra tools for symbolic summation. This, together with the combinatorial interpretation, allows us to realize some new determinantal relationships. From there, we are able to resolve all remaining open conjectures related to these determinants, including one from 2005 due to Lascoux and Krattenthaler.