Symmetry classes of spanning trees of Aztec diamonds and perfect matchings of odd squares with a unit hole

TL;DR

This paper explores the symmetry classes of spanning trees and perfect matchings in Aztec diamonds and related graphs, providing explicit formulas and combinatorial methods to count these structures.

Contribution

It introduces a new combinatorial approach to compute spanning trees and symmetry classes in Aztec diamonds and related graphs, revealing their linear squarishness property.

Findings

Explicit change of basis for adjacency matrices of Aztec-related graphs

Enumeration formulas for spanning trees and perfect matchings

Identification of linear squarishness property in various Aztec graph families

Abstract

We say that two graphs are similar if their adjacency matrices are similar matrices. We show that the square grid $G_n$ of order $n$ is similar to the disjoint union of two copies of the quartered Aztec diamond $QAD_{n-1}$ of order $n-1$ with the path $P_n^{(2)}$ on $n$ vertices having edge weights equal to~2. Our proof is based on an explicit change of basis in the vector space on which the adjacency matrix acts. The arguments verifying that this change of basis works are combinatorial. In particular, this allows computing the number of spanning trees of quartered Aztec diamonds. We present and analyze three more families of graphs that share the above described ``linear squarishness'' property of square grids: odd Aztec diamonds, mixed Aztec diamonds, and Aztec pillowcases--graphs obtained from two copies of an Aztec diamond by identifying the corresponding vertices on their convex hulls. We apply the above results to enumerate all the symmetry classes of spanning trees of the even Aztec diamonds, and all the symmetry classes not involving rotations of the spanning trees of odd and mixed Aztec diamonds. We also enumerate all but the base case of the symmetry classes of perfect matchings of odd square grids with the central vertex removed. In addition, we obtain a product formula for the number of spanning trees of Aztec pillowcases.