On the Number of Disjoint Pairs of S-permutation Matrices

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 $g$, $R_g \cap C_g =\emptyset$, $|R_g|=|C_g|=n$.

💡 Research Summary

The paper addresses a fundamental combinatorial problem that arises in the theory of Sudoku and related constraint‑based puzzles: counting the number of disjoint (mutually non‑overlapping) pairs of S‑permutation matrices of size (n^{2}\times n^{2}). An S‑permutation matrix is a binary matrix in which each row, each column, and each (n\times n) sub‑block contains exactly one entry equal to 1. Such matrices are the algebraic representation of the building blocks of Sudoku solutions. While earlier works by Geir Dahl introduced the concept of S‑permutation matrices and Roberto Fontana employed them in a constructive algorithm for generating Sudoku grids, neither provided a general formula for the total number of disjoint pairs as a function of the integer (n).

The author’s key insight is to translate the problem into the language of bipartite graph theory. For a fixed (n), consider two disjoint vertex sets (R_{g}) and (C_{g}), each of cardinality (n), representing the row‑blocks and column‑blocks of the matrix. An edge ((r,c)\in E_{g}) indicates that the cell at the intersection of row‑block (r) and column‑block (c) may host a 1. In this representation a single S‑permutation matrix corresponds to a perfect matching in the bipartite graph (g=\langle R_{g}\cup C_{g},E_{g}\rangle). Two matrices are disjoint precisely when the edge sets of their corresponding matchings are disjoint. Consequently, counting disjoint pairs reduces to enumerating ordered pairs of perfect matchings whose edge sets do not intersect.

To solve this enumeration, the paper proceeds in three stages. First, it classifies all possible bipartite graphs of the described type up to isomorphism. The classification depends only on the degree sequence ({d_{v}}{v\in V(g)}) and on the connectivity pattern of the graph. Second, for a given graph (g) with degree sequence ({d{v}}), the number of perfect matchings it supports is derived using rook‑polynomial techniques and the inclusion–exclusion principle. The result is a compact expression

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