Proof of Blum's conjecture on hexagonal dungeons

Mihai Ciucu; Tri Lai

doi:10.1016/j.jcta.2014.03.008

Proof of Blum's conjecture on hexagonal dungeons

Mihai Ciucu, Tri Lai

Research output: Contribution to journal › Article › peer-review

16 Scopus citations

Abstract

Matt Blum conjectured that the number of tilings of the hexagonal dungeon of sides a, 2. a, b, a, 2a, b (where b ≥ 2a) is 13^2a214⌊a²/2⌋ (Propp, 1999 [4]). In this paper we present a proof for this conjecture using Kuo's Graphical Condensation Theorem (Kuo, 2004 [2]).

Original language	English (US)
Pages (from-to)	273-305
Number of pages	33
Journal	Journal of Combinatorial Theory. Series A
Volume	125
Issue number	1
DOIs	https://doi.org/10.1016/j.jcta.2014.03.008
State	Published - Jul 2014
Externally published	Yes

Keywords

Aztec dungeons
Dual graph
Graphical condensation
Hexagonal dungeons
Perfect matchings
Tilings

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10.1016/j.jcta.2014.03.008

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abstract = "Matt Blum conjectured that the number of tilings of the hexagonal dungeon of sides a, 2. a, b, a, 2a, b (where b ≥ 2a) is 132a214⌊a2/2⌋ (Propp, 1999 [4]). In this paper we present a proof for this conjecture using Kuo's Graphical Condensation Theorem (Kuo, 2004 [2]).",

keywords = "Aztec dungeons, Dual graph, Graphical condensation, Hexagonal dungeons, Perfect matchings, Tilings",

author = "Mihai Ciucu and Tri Lai",

year = "2014",

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AU - Lai, Tri

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AB - Matt Blum conjectured that the number of tilings of the hexagonal dungeon of sides a, 2. a, b, a, 2a, b (where b ≥ 2a) is 132a214⌊a2/2⌋ (Propp, 1999 [4]). In this paper we present a proof for this conjecture using Kuo's Graphical Condensation Theorem (Kuo, 2004 [2]).

KW - Aztec dungeons

KW - Dual graph

KW - Graphical condensation

KW - Hexagonal dungeons

KW - Perfect matchings

KW - Tilings

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JF - Journal of Combinatorial Theory. Series A

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Proof of Blum's conjecture on hexagonal dungeons

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Keywords

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