On a Class of Binary Matrices

TL;DR

This paper investigates the structure and enumeration of square binary matrices with fixed row and column sums, establishing new combinatorial results and providing a novel proof of a known theorem, along with a classification of specific matrix subsets.

Contribution

It introduces a new connection between the counts of certain binary matrices and offers a novel proof of the Good-Grook theorem, including a classification of matrices with three 1's per row and column.

Findings

Established a relationship between the number of matrices with fixed row and column sums and those with a 1 in the lower right corner.

Provided a new proof of the Good-Grook theorem.

Classified matrices with three 1's in each row and column and a 1 in the lower right corner.

Abstract

The paper studies the set of all square binary matrices containing an exact number of 1's in each rows and in each column. A connection is established between the cardinal number of this set and the cardinal number of its subset of matrices containing 1 in the lower right corner. With the help of this result a new proof is advanced of the I. Good and J. Grook theorem. In connection with the firs result a classification has also been made of square binary matrices containing three 1's in each row and column and 1 in the lower right corner.