On the probability of two randomly generated S-permutation matrices to be disjoint

TL;DR

This paper investigates the probability that two randomly generated S-permutation matrices are disjoint, providing formulas for counting disjoint pairs and analyzing specific cases for small matrix sizes.

Contribution

It introduces a formula for counting disjoint pairs of S-permutation matrices and derives the probability of disjointness, including a new proof of a known result.

Findings

Lower bounds for disjoint S-permutation matrices are established.

A formula for counting disjoint pairs is proven.

Probability of disjointness is explicitly calculated for small cases.

Abstract

The concept of S-permutation matrix is considered in this paper. It defines when two binary matrices are disjoint. For an arbitrary $n^2 \times n^2$ S-permutation matrix, a lower band of the number of all disjoint with it S-permutation matrices is found. A formula for counting a lower band of the number of all disjoint pairs of $n^2 \times n^2$ S-permutation matrices is formulated and proven. As a consequence, a lower band of the probability of two randomly generated S-permutation matrices to be disjoint is found. In particular, a different proof of a known assertion is obtained in the work. The cases when $n=2$ and $n=3$ are discussed in detail.