  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, 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.
