Noncrossing partitions, noncrossing graphs, and q-permanental equations

We first prove a few simple results to illustrate some algebraic combinatorial features of the q-permanent. This is followed by a characterization of a noncrossing permutation in terms of the numbers of inversions of its cycles. Then we use a family of derivative formulas for the q-permanent of a square matrix A to characterize several structures of noncrossing kind. Each such formula f characterizes a set D-f of digraphs, in the sense that D is an element of D-f if f is valid for all matrices A with digraph D. In this way we characterize, among others, digraphs with non crossing permutation subdigraphs, noncrossing graphs, non crossing forests. We use the derivative formulas to prove two particular cases of a conjecture on the q-monotonicity of the q-permanent of a Hermitian positive definite matrix. (C) 2017 Elsevier Inc. All rights reserved.