Tilings of Benzels via Generalized Compression
Colin Defant, Leigh Foster, Rupert Li, James Gary Propp, Benjamin P. Young · SIAM Journal on Discrete Mathematics · 2025
Abstract. Defant, Li, Propp, and Young recently resolved two enumerative conjectures of Propp concerning the tilings of regions in the hexagonal grid called benzels using two types of prototiles called stones and bones (with varying constraints on allowed orientations of the tiles). Their primary tool, a bijection called compression that converts certain [Formula: see text]-ribbon tilings to [Formula: see text]-ribbon tilings, allowed them to reduce their problems to the enumeration of dimers (i.e., perfect matchings) of certain graphs. We present a generalized version of compression that no longer relies on the perspective of partitions and skew shapes. Using this strengthened tool, we resolve three more of Propp’s conjectures and recast several others as problems about perfect matchings.