The $\mu$-permanent, a new graph labeling, and a known integer sequence

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, motivated by this notion, we discuss a new graph labeling for trees whose matrices satisfy certain $\mu$-permanental identities. We relate the number of labelings of a path with a known integer sequence. Several examples are provided.