Ratio of tiling generating functions of semi-hexagons and quartered hexagons with dents

TL;DR

This paper studies the tiling generating functions of semi-hexagons and quartered hexagons with dents, revealing that modifications in width affect these functions by simple multiplicative factors, despite the absence of straightforward formulas.

Contribution

It demonstrates that changes in the regions' width lead to simple multiplicative adjustments in tiling generating functions, even without explicit product formulas.

Findings

Modification in width results in a simple multiplicative factor

No simple product formulas generally exist for these regions

Tiling generating functions are robust to certain geometric changes

Abstract

We consider the tiling generating functions of semi-hexagons and quartered hexagons with dents on their sides. In general, there are no simple product formulas for these generating functions. However, we show that the modification in the regions' width changes the tiling generating functions by only a simple multiplicative factor.