The $\mu$-permanent revisited

TL;DR

This paper reviews and extends the properties of the $$-permanent, a polynomial generalization of the permanent involving permutation inversions, discussing conjectures and correcting previous results.

Contribution

It revisits lesser-known properties of the $$-permanent, extends determinantal conjectures, and corrects earlier inaccuracies.

Findings

Discusses properties of the $$-permanent

Extends determinantal conjectures to the polynomial

Provides corrections to previous work

Abstract

Let $A=(a_{ij})$ be an $n$-by-$n$ matrix. For any real number $\mu$, we define the polynomial $$P_\mu(A)=\sum_{\sigma\in S_n} a_{1\sigma(1)}\cdots a_{n\sigma(n)}\,\mu^{\ell(\sigma)}\; ,$$ as the $\mu$-permanent of $A$, where $\ell(\sigma)$ is the number of inversions of the permutation $\sigma$ in the symmetric group $S_n$. In this note, we review several less known results of the $\mu$-permanent, recalling some of its interesting properties. Some determinantal conjectures are considered and extended to that polynomial. A correction to a previous note is presented as well.