An update on a few permanent conjectures

TL;DR

This paper reviews and updates several well-known conjectures related to matrix permanents, discussing recent developments, open questions, and new research directions for a general mathematical audience.

Contribution

It provides a comprehensive overview of multiple conjectures on matrix permanents, including recent resolutions and open problems, enhancing understanding in the field.

Findings

Some conjectures recently settled negatively

Identification of open conjectures and new questions

Summary of recent progress on permanent-related conjectures

Abstract

We review and update on a few conjectures concerning matrix permanent that are easily stated, understood, and accessible to general math audience. They are: Soules permanent-on-top conjecture${}^\dagger$, Lieb permanent dominance conjecture, Bapat and Sunder conjecture${}^\dagger$ on Hadamard product and diagonal entries, Chollet conjecture on Hadamard product, Marcus conjecture on permanent of permanents, and several other conjectures. Some of these conjectures are recently settled; some are still open. We also raise a few new questions for future study. (${}^\dagger$conjectures have been recently settled negatively.)