Proof of Blum's conjecture on hexagonal dungeons

Abstract

Matt Blum conjectured that the number of tilings of the Hexagonal Dungeon of sides $a,2a,b,a,2a,b$ (where $b\ge 2a$) is $13^{2a^2}14^{\lfloor \frac{a^2}{2}\rfloor }$ (J. Propp, New Perspectives in Geometric Combinatorics, Cambridge University Press, 1999). In this paper we present a proof for this conjecture using Kuo's Graphical Condensation Theorem (E. Kuo, Applications of Graphical Condensation for Enumerating Matchings and Tilings, Theoretical Computer Science, 2004).

30 pages