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Antichains of (0, 1)-matrices through inversions

Mohammad Ghebleh

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~$k$.

Antichains of (0, 1)-matrices through inversions

Abstract

An inversion in a matrix of zeros and ones consists of two entries both of which equal , and one of which is located to the top-right of the other. It is known that in the class of --matrices with row sum vector and column sum vector , the number of inversions in a matrix is monotonic with respect to the secondary Bruhat order. Hence any two matrices in the same class 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 , the class of all binary matrices with common row and column sum~. A product construction of antichains in the Bruhat order of is given. This product construction is applied in finding antichains in the Bruhat order of the class of square --matrices of order and common row and column sum~.
Paper Structure (4 sections, 8 theorems, 15 equations, 2 figures, 1 table)

This paper contains 4 sections, 8 theorems, 15 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Antichains in the Bruhat order of $\mathcal{A}(n,2)$
  3. A product construction
  4. Concluding remarks

Key Result

Lemma 1

Let $n\geqslant2$ and $t\geqslant0$ be integers. Then the set $\nu^{-1}(t)$ of all matrices $A\in\mathcal{A}(n,2)$ with $\nu(A)=t$ is an antichain in the Bruhat order of $\mathcal{A}(n,2)$.

Figures (2)

  • Figure 1: Histograms of the number of inversions in matrices in $\mathcal{A}(4,2)$ (top) and $\mathcal{A}(6,2)$ (bottom).
  • Figure 2: The two matrices $A_C$ and $A_C'$ constructed from $C$ in the proof of Theorem \ref{['thm:AcOdd']}.

Theorems & Definitions (14)

  • Lemma 1
  • Corollary 2
  • Lemma 3
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • Theorem 6
  • proof
  • Corollary 7
  • ...and 4 more