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

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.

💡 Research Summary

The paper investigates a special class of binary matrices called S‑permutation matrices, which are exactly the matrix representations of Sudoku solutions. An n² × n² S‑permutation matrix contains a single ‘1’ in each row, each column, and each n × n sub‑grid (block). Two such matrices A and B are defined to be disjoint when they never place a ‘1’ in the same cell, i.e., the entry‑wise product A ∘ B is the zero matrix. This notion captures the idea that two Sudoku solutions do not conflict with each other and is the cornerstone of the paper’s probabilistic analysis.

The first major contribution is a lower bound on the number of S‑permutation matrices that are disjoint from a given matrix M. The authors decompose M into its n² blocks and observe that, inside each block, the positions occupied by M’s ‘1’s must be avoided while still satisfying the row‑ and column‑permutation constraints. This avoidance problem is equivalent to finding a perfect matching in a bipartite graph whose vertices correspond to the rows and columns of the block and whose edges correspond to admissible cells. The number of admissible matchings for the k‑th block is shown to be

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