Domino Tilings of Aztec Octagons

Hyunggi Kim; Sangyop Lee; Seungsang Oh

doi:10.1007/s00373-023-02645-9

Domino Tilings of Aztec Octagons

Hyunggi Kim, Sangyop Lee, Seungsang Oh

Department of Mathematics

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

Abstract

Considerable energy has been devoted to understanding domino tilings: for example, Elkies, Kuperberg, Larsen and Propp proved the Aztec diamond theorem, which states that the number of domino tilings for the Aztec diamond of order n is equal to 2 ⁿ⁽ⁿ⁺¹⁾^/², and the authors recently counted the number of domino tilings for augmented Aztec rectangles and their chains by using Delannoy paths. In this paper, we count domino tilings for two new shapes of regions, bounded augmented Aztec rectangles and Aztec octagons by constructing a bijection between domino tilings for these regions and the associated generalized Motzkin paths.

Original language	English
Article number	45
Journal	Graphs and Combinatorics
Volume	39
Issue number	3
DOIs	https://doi.org/10.1007/s00373-023-02645-9
Publication status	Published - 2023 Jun

Keywords

Aztec diamond
Delannoy path
Domino tiling
Perfect matching

ASJC Scopus subject areas

Theoretical Computer Science
Discrete Mathematics and Combinatorics

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10.1007/s00373-023-02645-9

Cite this

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abstract = "Considerable energy has been devoted to understanding domino tilings: for example, Elkies, Kuperberg, Larsen and Propp proved the Aztec diamond theorem, which states that the number of domino tilings for the Aztec diamond of order n is equal to 2 n(n+1)/2, and the authors recently counted the number of domino tilings for augmented Aztec rectangles and their chains by using Delannoy paths. In this paper, we count domino tilings for two new shapes of regions, bounded augmented Aztec rectangles and Aztec octagons by constructing a bijection between domino tilings for these regions and the associated generalized Motzkin paths.",

keywords = "Aztec diamond, Delannoy path, Domino tiling, Perfect matching",

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AB - Considerable energy has been devoted to understanding domino tilings: for example, Elkies, Kuperberg, Larsen and Propp proved the Aztec diamond theorem, which states that the number of domino tilings for the Aztec diamond of order n is equal to 2 n(n+1)/2, and the authors recently counted the number of domino tilings for augmented Aztec rectangles and their chains by using Delannoy paths. In this paper, we count domino tilings for two new shapes of regions, bounded augmented Aztec rectangles and Aztec octagons by constructing a bijection between domino tilings for these regions and the associated generalized Motzkin paths.

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Domino Tilings of Aztec Octagons

Abstract

Keywords

ASJC Scopus subject areas

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