Restricted lonesum matrices

TL;DR

This paper studies restricted lonesum matrices, which are uniquely reconstructible matrices with limitations on the number of identical row and column types, connecting combinatorics, number theory, and graph orientations.

Contribution

It introduces and analyzes a new class of restricted lonesum matrices, extending their combinatorial and number-theoretic properties.

Findings

Enumeration formulas for restricted matrices

Connections to poly-Bernoulli numbers and zeta values

Insights into combinatorial structures and graph orientations

Abstract

Lonesum matrices are matrices that are uniquely reconstructible from their row and column sum vectors. These matrices are enumerated by the poly-Bernoulli numbers that are related to the multiple zeta values and have a rich literature in number theory. Combinatorially, lonesum matrices are in bijection with many other combinatorial objects: several permutation classes, other matrix classes, acyclic orientations in graphs etc. Motivated of these facts, we study in this paper lonesum matrices with restriction on the number of columns and rows of the same type.