Calculation of the Number of all Pairs of Disjoint S-permutation Matrices

TL;DR

This paper derives a general formula for counting disjoint pairs of S-permutation matrices of size n^2 by n^2 using graph theory, providing a mathematical foundation for understanding their combinatorial properties.

Contribution

It introduces a novel formula for counting disjoint S-permutation matrix pairs based on bipartite graph characteristics, advancing combinatorial matrix theory.

Findings

Derived a formula for counting disjoint pairs of S-permutation matrices

Connected matrix counting to bipartite graph characteristics

Proved the formula using graph theory techniques

Abstract

The concept of S-permutation matrix is considered. A general formula for counting all disjoint pairs of $n^2 \times n^2$ S-permutation matrices as a function of the positive integer $n$ is formulated and proven in this paper. To do that, the graph theory techniques have been used. It has been shown that to count the number of disjoint pairs of $n^2 \times n^2$ S-permutation matrices, it is sufficient to obtain some numerical characteristics of all $n\times n$ bipartite graphs.