Domino Tiling Congruence Modulo 4

TL;DR

This paper investigates the congruence properties of domino tilings in symmetric regions, revealing new modulo 4 relationships that extend previous modulo 2 results and include specific cases like rectangles.

Contribution

It introduces a novel combinatorial approach to analyze domino tilings modulo 4, expanding the understanding of tiling congruences beyond earlier modulo 2 findings.

Findings

Number of domino tilings of symmetric regions depends on subregion tilings modulo 4

Rectangles of size k x 2k have tilings congruent to 1 mod 4

Extends previous results from modulo 2 to modulo 4

Abstract

The number of domino tilings of a region with reflective symmetry across a line is combinatorially shown to depend on the number of domino tilings of particular subregions, modulo 4. This expands upon previous congruency results for domino tilings, modulo 2, and leads to a variety of corollaries, including that the number of domino tilings of a k x 2k rectangle is congruent to 1 mod 4.