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Domino Tilings of Aztec Octagons

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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.

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Notes

  1. Remark that they miswrote ‘\(\sin \frac{(i + 1) m \pi }{n + 2} \, \sin \frac{(j + 1) m \pi }{n + 2}\)’ by ‘\(\sin \frac{i m \pi }{n + 2} \, \sin \frac{j m \pi }{n + 2}\)’ in [7].

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Authors and Affiliations

  1. Department of Mathematical Science, Seoul National University, Seoul, 08826, Korea

    Hyunggi Kim

  2. Department of Mathematics, Chung-Ang University, Seoul, 06974, Korea

    Sangyop Lee

  3. Department of Mathematics, Korea University, Seoul, 02841, Korea

    Seungsang Oh

Authors
  1. Hyunggi Kim
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  2. Sangyop Lee
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  3. Seungsang Oh
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Correspondence to Seungsang Oh.

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The second author was supported by the National Research Foundation of Korea Grant funded by the Korean Government (NRF-2020R1F1A1A01074716). The third author was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (NRF-2022R1F1A1064273).

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Kim, H., Lee, S. & Oh, S. Domino Tilings of Aztec Octagons. Graphs and Combinatorics 39, 45 (2023). https://doi.org/10.1007/s00373-023-02645-9

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  • DOI: https://doi.org/10.1007/s00373-023-02645-9

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