Determinant evaluations inspired by Di Francesco's determinant for twenty-vertex configurations

Christoph Koutschan, Christian Krattenthaler, Michael Schlosser

Abstract

In his work on the twenty vertex model, Di Francesco [Electron. J. Combin. 28(4) (2021), Paper No. 4.38] found a determinant formula for the number of configurations in a specific such model, and he conjectured a closed form product formula for the evaluation of this determinant. We prove this conjecture here. Moreover, we actually generalize this determinant evaluation to a one-parameter family of determinant evaluations, and we present many more determinant evaluations of similar type — some proved, some left open as conjectures.

Paper

The material on this webpage accompanies the article Determinant evaluations inspired by Di Francesco's determinant for twenty-vertex configurations by Christoph Koutschan, Christian Krattenthaler, and Michael Schlosser.

Supplementary electronic material

We provide a Mathematica notebook containing all computations that constitute the proofs of the determinant evaluations in our article. 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 articles Advanced computer algebra for determinants and A curious family of binomial determinants that count rhombus tilings of a holey hexagon 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 files into the same directory where the notebook is stored.

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