Spectrum of permanent's values and its extremal magnitudes in $\Lambda_n^3$ and $\Lambda_n(\alpha,\beta,\gamma)$

TL;DR

This paper investigates the extremal values of the permanent in specific classes of (0,1) matrices, providing algorithms and analyzing generalized matrix classes with fixed nonzero elements per row and column.

Contribution

It introduces a simple algorithm for upper bounds of the permanent in $ ext{Lambda}_n^3$ and explores extremal problems in generalized matrix classes with fixed nonzero entries.

Findings

Developed an algorithm for upper bounds of the permanent in $ ext{Lambda}_n^3$.

Analyzed extremal properties in generalized classes $ ext{Lambda}_n( ext{alpha}, ext{beta}, ext{gamma})$.

Abstract

Let $\Lambda_n^k$ denote the class of $(0,1)$ square matrices containing in each row and in each column exactly $k$ 1's. The minimal value of $k,$ for which the behavior of the permanent in $\Lambda_n^k$ is not quite studied, is $k=3.$ We give a simple algorithm for calculation upper magnitudes of permanent in $\Lambda_n^3$ and consider some extremal problems in a generalized class $\Lambda_n(\alpha,\beta,\gamma),$ the matrices of which contain in each row and in each column nonzero elements $\alpha,\beta,\gamma$ and $n-3$ zeros.