Milica Andelić, Carlos M. da Fonseca, António Pereira: The μ-permanent, a new graph labeling, and a known integer sequence, 255-262

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, 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.

Key Words: $\mu$-permanent, $q$-permanent, determinant, permanent, graph, tree, path, graph labeling, integer sequence, Mathematica

2010 Mathematics Subject Classification: Primary 15A15. Secondary 05C50, 05C78, 05C30, 68R10, 11B83