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

2016/09/14 by Milica Anđelić, Carlos M. da Fonseca, Anđelić, Milica +3
Mathematics · #05C30 #05C50 #05C78 #11B83 #15A15 #68R10 #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:05C30 #msc:05C50 #msc:05C78 #msc:11B83 #msc:15A15 #msc:68R10

paper · pdf · doi:10.48550/arxiv.1609.04208

The example for a $μ$-labeling on page 3 has been corrected

arxiv created 2016/09/14 · arxiv updated 2016/09/15

Abstract

Let A=(a_ij) be an n-by-n matrix. For any real number μ, we define the polynomial P_μ(A)=∑_{σ∈ Sn} a_1σ(1)⋯ a_nσ(n) μ^ℓ(σ) , as the μ-permanent of A, where ℓ(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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