Three product formulas for ratios of tiling counts of hexagons with collinear holes

TL;DR

This paper generalizes product formulas for tiling counts of hexagons with collinear holes, extending previous results and proving a recent conjecture about symmetric tilings.

Contribution

It introduces new product formulas for tilings of hexagons with arbitrary collinear holes and confirms a conjecture on symmetric tilings with a fern removed.

Findings

Derived explicit formulas for tiling ratios with collinear holes

Extended Ciucu's work to more general hexagon tilings

Proved a recent conjecture on symmetric tilings

Abstract

Rosengren found an explicit formula for a certain weighted enumeration of lozenge tilings of a hexagon with an arbitrary triangular hole. He pointed out that a certain ratio corresponding to two such regions has a nice product formula. In this paper, we generalize this to hexagons with arbitrary collinear holes. It turns out that, by using same approach, we can also generalize Ciucu's work on the number and the number of centrally symmetric tilings of a hexagon with a fern removed from its center. This proves a recent conjecture of Ciucu.