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

Matt Blum conjectured that the- number of tilings of a hexagonal dungeon with side-lengths a,2a,b,a,2a,b (for b≥2a) equals 132a214⌊a2∕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.