Domino tilings, nonintersecting lattice paths and subclasses of Koutschan-Krattenthaler-Schlosser determinants

TL;DR

This paper provides combinatorial interpretations for certain binomial determinants using domino tilings and lattice paths, proves two conjectural formulas, and advances understanding of these mathematical structures.

Contribution

It offers new combinatorial interpretations of specific determinants and proves two previously conjectural formulas using advanced algebraic techniques.

Findings

Combinatorial interpretations of determinants via domino tilings and lattice paths

Proof of two conjectural formulas for weighted enumerations

Extension of determinant evaluations to new combinatorial contexts

Abstract

Koutschan, Krattenthaler and Schlosser recently considered a family of binomial determinants. In this work, we give combinatorial interpretations of two subclasses of these determinants in terms of domino tilings and nonintersecting lattice paths, thereby partially answering a question of theirs. Furthermore, the determinant evaluations established by Koutschan, Krattenthaler and Schlosser produce many product formulas for our weighted enumerations of domino tilings and nonintersecting lattice paths. However, there are still two enumerations left corresponding to conjectural formulas made by the three. We hereby prove the two conjectures using the principle of holonomic Ansatz plus the approach of modular reduction for creative telescoping, and hence fill the gap.