A Determinantal Identity for the Permanent of a Rank 2 Matrix

TL;DR

This paper presents a new identity linking the permanent of a rank 2 matrix to determinants of its Hadamard powers, revealing potential broader connections between permanents and determinants.

Contribution

It introduces a novel determinantal identity for the permanent of rank 2 matrices, inspired by classical identities and combining symmetric function theory with linear algebra.

Findings

Established a new identity relating permanents and determinants for rank 2 matrices

Connected the identity to classical results of Carlitz and Levine

Suggested potential for more general identities linking permanents and determinants

Abstract

We prove an identity relating the permanent of a rank $2$ matrix and the determinants of its Hadamard powers. When viewed in the right way, the resulting formula looks strikingly similar to an identity of Carlitz and Levine, suggesting the possibility that these are actually special cases of some more general identity (or class of identities) connecting permanents and determinants. The proof combines some basic facts from the theory of symmetric functions with an application of a famous theorem of Binet and Cauchy in linear algebra.