On the number of mutually disjoint pairs of S-permutation matrices

TL;DR

This paper introduces a formula and an algorithm for counting and analyzing mutually disjoint pairs of S-permutation matrices, which are special permutation matrices with a block structure, using a novel factor-set approach.

Contribution

It provides a new formula and an algorithm for counting disjoint S-permutation matrix pairs based on a unique n-tuple representation of binary matrices.

Findings

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

Developed an algorithm utilizing a factor-set of binary matrices.

Represented binary matrices as n-tuples of integers for analysis.

Abstract

This work examines the concept of S-permutation matrices, namely $n^2 \times n^2$ permutation matrices containing a single 1 in each canonical $n \times n$ subsquare (block). The article suggests a formula for counting mutually disjoint pairs of $n^2 \times n^2$ S-permutation matrices in the general case by restricting this task to the problem of finding some numerical characteristics of the elements of specially defined for this purpose factor-set of the set of $n \times n$ binary matrices. The paper describe an algorithm that solves the main problem. To do that, every $n\times n$ binary matrix is represented uniquely as a n-tuple of integers.