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

Tri Lai

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

Combinatorics 2015-04-02 v4

Authors: Tri Lai

Abstract

Matt Blum conjectured that the number of tilings of a hexagonal dungeon with side-lengths $a,2a,b,a,2a,b$ (for $b\geq2a$ ) equals $13^{2a^2}14^{\lfloor a^2/2\rfloor}$ . 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.

Keywords

hexagonal dungeons tilings Blum's conjecture graphical condensation enumeration Kuo's method

Cite

@article{arxiv.1403.4481,
  title  = {Proof of a refinement of Blum's conjecture on hexagonal dungeons},
  author = {Tri Lai},
  journal= {arXiv preprint arXiv:1403.4481},
  year   = {2015}
}

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22 pages

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