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 Df of digraphs, in the sense that D∈Df iff f is valid for all matrices A with digraph D. In this way we characterize, among others, digraphs with noncrossing permutation subdigraphs, noncrossing graphs, noncrossing 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.