Enumeration of Hybrid Domino-Lozenge Tilings III: Centrally Symmetric Tilings

TL;DR

This paper investigates centrally symmetric tilings of regions on the square lattice with diagonals, providing new enumeration formulas and generalizing previous results on symmetric Aztec diamond tilings.

Contribution

It introduces a subgraph replacement method to derive simple product formulas for centrally symmetric tilings of generalized Aztec diamonds and quasi-hexagons.

Findings

Centrally symmetric tilings of generalized Aztec diamonds are counted by a simple product formula.

Derived a closed-form product formula for centrally symmetric tilings of quasi-hexagons.

Extended previous enumeration results to broader classes of symmetric tilings.

Abstract

We use the subgraph replacement method to investigate new properties of the tilings of regions on the square lattice with diagonals drawn in. In particular, we show that the centrally symmetric tilings of a generalization of the Aztec diamond are always enumerated by a simple product formula. This result generalizes the previous work of Ciucu (1997) and Yang (1992) about symmetric tilings of the Aztec diamond. We also use our method to prove a closed-form product formula for the number of centrally symmetric tilings of a quasi-hexagon.