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

TL;DR

The paper proves that for even $d$, every $d$-dimensional polystochastic matrix of order $4$ has a positive permanent, and for odd $d$ it fully characterizes zero-permanent examples as those equivalent to $\,\mathcal{M}_4^d$ or to matrices in the family $\,\mathcal{L}_4^d$. The authors develop a reduction to sesquialteral permutations, establish bitrade structure from unitrade analysis, and use block-permutation and tessellation techniques to tightly constrain possible zero-permanent configurations, culminating in a complete classification. The results resolve the $n=4$ case of the broader conjecture on permanents of polystochastic matrices and clarify the role of Latin hypercubes and related combinatorial constructions in multidimensional permanents.

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.

Theorems & Definitions (22)

• Conjecture 1: Wanless, Wan
• Conjecture 2: Taranenko, AAT16
• Proposition 1
• Lemma 1
• Theorem 1
• Lemma 2
• Lemma 3
• Theorem 2: AAT18, Theorem 5
• Lemma 4
• Proposition 2