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

Let $A=(a_{ij})$ be an $n$-by-$n$ matrix. For any real number $μ$, we define the polynomial $P_μ(A)=\sum_{σ\in S_n} a_{1σ(1)}\cdots a_{nσ(n)}\,μ^{\ell(σ)}\; ,$ as the $μ$-permanent of $A$, where $\ell(σ)$ is the number of inversions of the permutation $σ$ 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 $μ$-permanental identities. We relate the number of labelings of a path with a known integer sequence. Several examples are provided.

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