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.
Bibliographical noteFunding Information:
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).
© The Author(s), under exclusive licence to Springer Nature Japan KK, part of Springer Nature 2023.
- Aztec diamond
- Delannoy path
- Domino tiling
- Perfect matching
ASJC Scopus subject areas
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics