Non-P-recursiveness of numbers of matchings or linear chord diagrams with many crossings

The number con_n counts matchings X on {1,2,…,2n}, which are partitions into n two-element blocks, such that the crossing graph of X is connected. Similarly, cro_n counts matchings whose crossing graph has no isolated vertex. (If it has no edge, Catalan numbers arise.) We apply generating functions techniques and prove, using a more generally applicable criterion, that the sequences (con_n) and (cro_n) are not P-recursive. On the other hand, we show that the residues of con_n and cro_n modulo any fixed power of 2 can be determined P-recursively. We consider also the numbers sco_n of symmetric connected matchings. Unfortunately, their generating function satisfies a complicated differential equation which we cannot handle.