A Note on One-Hole Domino Tilings of Squares and Rectangles
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866918045132259328 |
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| author | Byun, Seok Hyun Goddard, Wayne |
| author_facet | Byun, Seok Hyun Goddard, Wayne |
| contents | We consider the number of domino tilings of an odd-by-odd rectangle that leave one hole. This problem is equivalent to the number of near-perfect matchings of the odd-by-odd rectangular grid. For any particular position of the vacancy on the $(2k+1)\times (2k+1)$ square grid, we show that the number of near-perfect matchings is a multiple of $2^k$, and from this follows a conjecture of Kong that the total number of near-perfect matchings is a multiple of $2^k$. We also determine the parity of the number of near-perfect matchings with a particular vacancy for the rectangle case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_05918 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Note on One-Hole Domino Tilings of Squares and Rectangles Byun, Seok Hyun Goddard, Wayne Combinatorics 05A15, 05A19 We consider the number of domino tilings of an odd-by-odd rectangle that leave one hole. This problem is equivalent to the number of near-perfect matchings of the odd-by-odd rectangular grid. For any particular position of the vacancy on the $(2k+1)\times (2k+1)$ square grid, we show that the number of near-perfect matchings is a multiple of $2^k$, and from this follows a conjecture of Kong that the total number of near-perfect matchings is a multiple of $2^k$. We also determine the parity of the number of near-perfect matchings with a particular vacancy for the rectangle case. |
| title | A Note on One-Hole Domino Tilings of Squares and Rectangles |
| topic | Combinatorics 05A15, 05A19 |
| url | https://arxiv.org/abs/2502.05918 |