# Lonesum decomposable matrices

## TL;DR

This paper introduces lonesum decomposable matrices, characterizes their properties, and provides a generating function for counting such matrices, along with congruence relations modulo primes.

## Contribution

It defines lonesum decomposable matrices, establishes their key properties, and derives a generating function for counting them, including congruence results.

## Key findings

- Derived a necessary and sufficient condition for lonesum decomposability.
- Provided a generating function for counting lonesum decomposable matrices.
- Proved congruences for the number of such matrices modulo primes.

## Abstract

A lonesum matrix is a $(0,1)$-matrix that is uniquely determined by its row and column sum vectors. In this paper, we introduce lonesum decomposable matrices and study their properties. We provide a necessary and sufficient condition for a matrix $A$ to be lonesum decomposable, and give a generating function for the number $D_k(m,n)$ of $m\times n$ lonesum decomposable matrices of order $k$. Moreover, by using this generating function we prove some congruences for $D_k(m,n)$ modulo a prime.