On the Number of Disjoint Pairs of S-permutation Matrices

TL;DR

This paper derives a formula for counting disjoint pairs of S-permutation matrices of size n^2 by using graph theory techniques to analyze bipartite graphs, advancing combinatorial understanding relevant to Sudoku matrix algorithms.

Contribution

It provides the first general formula for counting disjoint pairs of S-permutation matrices using bipartite graph characteristics.

Findings

Derived a formula for disjoint pairs count

Used bipartite graph analysis for combinatorial enumeration

Enhanced understanding of Sudoku matrix structure

Abstract

In [Journal of Statistical Planning and Inference (141) (2011) 3697-3704], Roberto Fontana offers an algorithm for obtaining Sudoku matrices. Introduced by Geir Dahl concept disjoint pairs of S-permutation matrices [Linear Algebra and its Applications (430) (2009) 2457-2463] is used in this algorithm. Analyzing the works of G. Dahl and R. Fontana, the question of finding a general formula for counting disjoint pairs of $n^2 \times n^2$ S-permutation matrices as a function of the integer $n$ naturally arises. This is an interesting combinatorial problem that deserves its consideration. The present work solves this problem. 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 the set of all bipartite graphs of the type $g=<R_g \cup C_g, E_g>$, where $V=R_g \cup C_g$ is the set of vertices, and $E_g$ is the set of edges of the graph $g$, $R_g \cap C_g =\emptyset$, $|R_g|=|C_g|=n$.