Linear preservers for the q-permanent, cycle q-permanent expansions, and positive crossings in digraphs

The q-permanent linear preservers are described. We give several expansion formulas for the q-permanent of a square matrix, based on the cycle factorization of permutations. Some of these formulas are valid for all matrices, but others are not; for each such formula Φ we determine all digraphs D such that Φ holds for all matrices with digraph D. Proof techniques are based on combinatorial results, relating the length (number of inversions) of a permutation, the lengths of its cycles, and a delicate counting of crossings, jumps, and arc-under-arc relations in digraphs. We get new algebraic characterizations of noncrossing [acyclic] graphs.