Domino Tilings of Cruciform Regions

TL;DR

This paper generalizes a known domino tiling formula for cruciform regions, extending previous work on Aztec triangles and providing a generating-function approach.

Contribution

It introduces a generating-function formula for domino tilings of cruciform regions, expanding upon Ciucu's product formula.

Findings

Derived a generating-function formula for cruciform tilings

Extended Ciucu's tiling formula to a broader class of regions

Connected tiling enumeration to generating functions

Abstract

P. Di Francesco first introduced the "Aztec triangle" in his study of the relationship between the twenty-vertex model and domino tilings. He conjectured an exact formula for the number of tilings of the Aztec triangle, and it has since been proved by several authors. In an attempt to prove the conjecture, M. Ciucu showed that the tiling number of the Aztec triangle divides the tiling number of a new region called the "cruciform region," a superposition of two Aztec rectangles. Ciucu proved that the number of domino tilings of a cruciform region is given by a simple product formula. In this paper, we generalize Ciucu's tiling formula by providing a generating-function formula for the cruciform region.