New permanent approximation inequalities via identities

TL;DR

This paper introduces new inequalities that provide tighter bounds on the difference between a normalized permanent of a matrix and the product of column means, improving previous results through novel identities and generalizations.

Contribution

It presents new upper bounds for permanent approximations using innovative identities, extending classical results and including second-order inequalities.

Findings

Improved bounds for permanent approximation errors

New identities generalizing Dougall's classical results

Inclusion of second-order approximation inequalities

Abstract

The aim of this paper is to present new upper bounds for the distance between a properly normalized permanent of a rectangular complex matrix and the product of the arithmetic means of the entries of its columns. It turns out that the bounds improve on those from earlier work. Our proofs are based on some new identities for the above-mentioned differences and also for related expressions for matrices over a rational associative commutative unital algebra. Some of our identities are generalizations of results in Dougall (Proceedings of the Edinburgh Mathematical Society, 24, 61-77, 1905). Second order results are also included.