Proof of a refinement of Blum's conjecture on hexagonal dungeons

Tri Lai

paper

Proof of a refinement of Blum's conjecture on hexagonal dungeons

arXiv:1403.4481

Abstract

Matt Blum conjectured that the number of tilings of a hexagonal dungeon with side-lengths $a, 2 a, b, a, 2 a, b$ (for $b \geq 2 a$ ) equals $1 3^{2 a^{2}} 1 4^{⌊ a^{2} /2 ⌋}$ . Ciucu and the author of the present paper proved the conjecture by using Kuo's graphical condensation method. In this paper, we investigate a 3-parameter refinement of the conjecture and its application to enumeration of tilings of several new types of the hexagonal dungeons.

22 pages