Antichains of (0, 1)-matrices through inversions
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909084868935680 |
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| author | Ghebleh, Mohammad |
| author_facet | Ghebleh, Mohammad |
| contents | An inversion in a matrix of zeros and ones consists of two entries both of which equal $1$, and one of which is located to the top-right of the other. It is known that in the class $\mathcal{A}(R,S)$ of $(0,1)$--matrices with row sum vector $R$ and column sum vector $S$, the number of inversions in a matrix is monotonic with respect to the secondary Bruhat order. Hence any two matrices in the same class $\mathcal{A}(R,S)$ having the same number of inversions, are incomparable in the secondary Bruhat order. We use this fact to construct antichains in the Bruhat order of $\mathcal{A}(n,2)$, the class of all $n\times n$ binary matrices with common row and column sum~$2$.
A product construction of antichains in the Bruhat order of $\mathcal{A}(R,S)$ is given. This product construction is applied in finding antichains in the Bruhat order of the class $\mathcal{A}(2k,k)$ of square $(0,1)$--matrices of order $2k$ and common row and column sum~$k$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_15437 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Antichains of (0, 1)-matrices through inversions Ghebleh, Mohammad Combinatorics 05B20, 06A07, 15A36 An inversion in a matrix of zeros and ones consists of two entries both of which equal $1$, and one of which is located to the top-right of the other. It is known that in the class $\mathcal{A}(R,S)$ of $(0,1)$--matrices with row sum vector $R$ and column sum vector $S$, the number of inversions in a matrix is monotonic with respect to the secondary Bruhat order. Hence any two matrices in the same class $\mathcal{A}(R,S)$ having the same number of inversions, are incomparable in the secondary Bruhat order. We use this fact to construct antichains in the Bruhat order of $\mathcal{A}(n,2)$, the class of all $n\times n$ binary matrices with common row and column sum~$2$. A product construction of antichains in the Bruhat order of $\mathcal{A}(R,S)$ is given. This product construction is applied in finding antichains in the Bruhat order of the class $\mathcal{A}(2k,k)$ of square $(0,1)$--matrices of order $2k$ and common row and column sum~$k$. |
| title | Antichains of (0, 1)-matrices through inversions |
| topic | Combinatorics 05B20, 06A07, 15A36 |
| url | https://arxiv.org/abs/2401.15437 |