On a conjecture about the µ-permanent

Abstract

Let A=(aij) be an n-by-nmatrix. For any real µ, define the polynomial Pµ(A)=Σ (σ E Sn) α1 σ(1) . . . αnσ(n)µ l(σ) where l (s) is the number of inversions of the permutation s in the symmetric group Sn. We prove that Pµ (A)is a strictly increasing function of µ ? [-1,1], for a Hermitian positive definite nondiagonal matrix A, whose graph is a tree.