The $\mu$-permanent revisited

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.

This manuscript is largely based on the talk "On a generalization of the determinant of a matrix" that the author gave in Washington \& Lee University, Lexington, VA, USA, on July 19, 2012. It was submitted to Linear and Multilinear Algebra on September 20, 2015. A reduced form will be published as a Letter to the Editor in the same journal