Characterization of polystochastic matrices of order $4$ with zero permanent

TL;DR

This paper investigates the properties of polystochastic matrices of order 4, proving positivity of their permanent for even dimensions and characterizing those with zero permanent for odd dimensions.

Contribution

It establishes the positivity of the permanent for even-dimensional polystochastic matrices of order 4 and fully characterizes zero-permanent matrices in odd dimensions.

Findings

Permanent is positive for even d

Complete characterization of zero permanent matrices for odd d

Advances understanding of multidimensional stochastic matrices

Abstract

A multidimensional nonnegative matrix is called polystochastic if the sum of its entries over each line is equal to $1$. The permanent of a multidimensional matrix is the sum of products of entries over all diagonals. We prove that if $d$ is even, then the permanent of a $d$-dimensional polystochastic matrix of order $4$ is positive, and for odd $d$, we give a complete characterization of $d$-dimensional polystochastic matrices with zero permanent.