A remark on an equivalence of two versions of Polya's permanent problem

TL;DR

This paper corrects a flaw in the known equivalence of two versions of Polya's permanent problem by providing a new characterization for when a matrix's determinant equals its permanent, including conditions for their equality across powers.

Contribution

It offers a corrected and extended characterization of when the determinant equals the permanent for nonnegative matrices, addressing a flaw in previous equivalence claims.

Findings

Provides necessary and sufficient conditions for determinant and permanent equality.

Characterizes when determinant equals permanent for powers of matrices.

Corrects a flaw in the previous equivalence of two versions of Polya's problem.

Abstract

In 2004, some equivalent versions of Polya's permanent problem were listed in 24 versions. However, there is a flaw on the theorem that affirms an equivalence of version 11 and 12. In order to correct the slip, we provide a characterization when the determinant and permanent of a given nonnegative matrix are equal. Moreover, the new characterization yields necessary and sufficient conditions for the equality of the determinant and permanent of powers of a given nonnegative matrix.