On some classes of binary matrices

TL;DR

This paper studies special classes of binary matrices with fixed row and column sums, focusing on lexicographically sorted matrices and revealing connections to Fibonacci numbers.

Contribution

It introduces and analyzes classes of binary matrices with fixed row and column sums, especially those with lexicographically sorted rows and columns, and explores their combinatorial properties.

Findings

Identifies properties of lexicographically sorted binary matrices.

Establishes relationships between matrix counts and Fibonacci numbers.

Provides insights into special cases of fixed-sum binary matrices.

Abstract

The work considers the set $\Lambda_n^k$ of all $n\times n$ binary matrices having the same number of $k$ units in each row and each column. The article specifically focuses on the matrices whose rows and columns are sorted lexicographically. We examine some particular cases and special properties of this matrices. Finally, we demonstrate the relationship between the Fibonacci numbers and the cardinality of two classes of $\Lambda_n^k$-matrices with lexicographically sorted rows and columns.