Domino tilings of the expanded Aztec diamond

TL;DR

This paper introduces a recurrence relation approach to exactly count domino tilings of the expanded Aztec diamond, a generalized lattice with variable long columns and rows, extending previous enumeration methods.

Contribution

It develops a novel state matrix recursion algorithm to derive exact counts for domino tilings of the expanded Aztec diamond, generalizing classical results.

Findings

Derived recurrence relations for domino tilings

Extended enumeration techniques to generalized Aztec diamonds

Provided exact tiling counts for complex lattice structures

Abstract

The expanded Aztec diamond is a generalized version of the Aztec diamond, with an arbitrary number of long columns and long rows in the middle. In this paper, we count the number of domino tilings of the expanded Aztec diamond. The exact number of domino tilings is given by recurrence relations of state matrices by virtue of the state matrix recursion algorithm, recently developed by the author to solve various two-dimensional regular lattice model enumeration problems.