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arXiv · 2401.15437

Antichains of (0, 1)-matrices through inversions

Mohammad Ghebleh
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Abstract

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~$k$.

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Mohammad Ghebleh. 2024-01-27. Antichains of (0, 1)-matrices through inversions. https://doi.org/10.1016/j.laa.2014.06.021

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