A Note on One-Hole Domino Tilings of Squares and Rectangles

TL;DR

This paper investigates the number of domino tilings with one hole in odd-by-odd rectangles, establishing divisibility properties and parity results for near-perfect matchings, thus contributing to combinatorial tiling theory.

Contribution

It proves that the number of near-perfect matchings with a fixed vacancy is divisible by a power of two and confirms Kong's conjecture on total matchings.

Findings

Number of near-perfect matchings is divisible by 2^k for a fixed vacancy.

Total near-perfect matchings are divisible by 2^k, supporting Kong's conjecture.

Determined the parity of near-perfect matchings with a specific vacancy.

Abstract

We consider the number of domino tilings of an odd-by-odd rectangle that leave one hole. This problem is equivalent to the number of near-perfect matchings of the odd-by-odd rectangular grid. For any particular position of the vacancy on the $(2k+1)\times (2k+1)$ square grid, we show that the number of near-perfect matchings is a multiple of $2^k$, and from this follows a conjecture of Kong that the total number of near-perfect matchings is a multiple of $2^k$. We also determine the parity of the number of near-perfect matchings with a particular vacancy for the rectangle case.