Off-diagonally symmetric alternating sign matrices
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908281130188800 |
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| author | Kumari, Nishu |
| author_facet | Kumari, Nishu |
| contents | A diagonally symmetric alternating sign matrix (DSASM) is a symmetric matrix with entries $-1$, $0$ and $1$, where the nonzero entries alternate in sign along each row and column, and the sum of the entries in each row and column equals $1$. An off-diagonally symmetric alternating sign matrix (OSASM) is a DSASM, where the number of nonzero diagonal entries is 0 for even-order matrices and 1 for odd-order matrices. Kuperberg (Ann. Math., 2002) studied even-order OSASMs and derived a product formula for counting the number of OSASMs of any fixed even order. In this work, we provide a product formula for the number of odd-order OSASMs of any fixed order. Additionally, we present an algebraic proof of a symmetry property for even-order OSASMs. This resolves all the three conjectures of Behrend, Fischer, and Koutschan (arXiv, 2023) regarding the exact enumeration of OSASMs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_18685 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Off-diagonally symmetric alternating sign matrices Kumari, Nishu Combinatorics 05B20, 15B35, 05E10, 05E05 A diagonally symmetric alternating sign matrix (DSASM) is a symmetric matrix with entries $-1$, $0$ and $1$, where the nonzero entries alternate in sign along each row and column, and the sum of the entries in each row and column equals $1$. An off-diagonally symmetric alternating sign matrix (OSASM) is a DSASM, where the number of nonzero diagonal entries is 0 for even-order matrices and 1 for odd-order matrices. Kuperberg (Ann. Math., 2002) studied even-order OSASMs and derived a product formula for counting the number of OSASMs of any fixed even order. In this work, we provide a product formula for the number of odd-order OSASMs of any fixed order. Additionally, we present an algebraic proof of a symmetry property for even-order OSASMs. This resolves all the three conjectures of Behrend, Fischer, and Koutschan (arXiv, 2023) regarding the exact enumeration of OSASMs. |
| title | Off-diagonally symmetric alternating sign matrices |
| topic | Combinatorics 05B20, 15B35, 05E10, 05E05 |
| url | https://arxiv.org/abs/2503.18685 |