Laurent biorthogonal polynomials, q-Narayana polynomials and domino tilings of the Aztec diamonds

TL;DR

This paper introduces Laurent biorthogonal polynomials linked to q-Narayana polynomials, providing a combinatorial approach to evaluate a T"oplitz determinant and offering a new proof of the Aztec diamond theorem through Schr"oder paths.

Contribution

It develops a novel combinatorial interpretation using Laurent biorthogonal polynomials for evaluating determinants related to q-Narayana polynomials and offers a new proof of the Aztec diamond theorem.

Findings

Evaluation of a T"oplitz determinant via Laurent biorthogonal polynomials

Combinatorial interpretation using Schr"oder paths

New proof of the Aztec diamond theorem

Abstract

A T\"oplitz determinant whose entries are described by a q-analogue of the Narayana polynomials is evaluated by means of Laurent biorthogonal polynomials which allow of a combinatorial interpretation in terms of Schr\"oder paths. As an application, a new proof is given to the Aztec diamond theorem by Elkies, Kuperberg, Larsen and Propp concerning domino tilings of the Aztec diamonds. The proof is based on the correspondence with non-intersecting Schr\"oder paths developed by Eu and Fu.