The $\mu$-permanent revisited

Let $A=(a_{ij})$ be an $n$-by-$n$ matrix. For any real number $\mu$, we define the polynomial $$P_{\mu }(A)=\sum _{\sigma \in S_n}{a}_{1\sigma (1)}\cdots a_{n\sigma (n)}{\mu }^{\mathrm{\ell }(\sigma )},$$ as the $\mu$-permanent of $A$, where $\mathrm{\ell }(\sigma )$ 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.