A Generalization of Aztec Dragons

TL;DR

This paper generalizes Aztec dragons to new 6-sided regions and proves their tilings are always counted by powers of 2 and 3 using Kuo's graphical condensation method.

Contribution

It introduces two new families of regions extending Aztec dragons and establishes their tiling counts as powers of 2 and 3, expanding understanding of lattice tilings.

Findings

Tilings of the new regions are always powers of 2 and 3.

The paper extends Aztec dragon enumeration to broader families.

Uses Kuo's graphical condensation method for proofs.

Abstract

Aztec dragons are lattice regions first introduced by James Propp, which have the number of tilings given by a power of $2$. This family of regions has been investigated further by a number of authors. In this paper, we consider a generalization of the Aztec dragons to two new families of $6$-sided regions. By using Kuo's graphical condensation method, we prove that the tilings of the new regions are always enumerated by powers of $2$ and $3$.