On Non P-recursiveness of Numbers of Matchings (linear Chord Diagrams) with Many Crossings

The number conn counts matchings X on f1; 2; : : : ; 2ng, which are partitions into n two-element blocks, such that the crossing graph of X is connected. Similarly, cron counts matchings whose crossing graph has no isolated vertex. (If it has no edge, Catalan numbers arise.) We prove, using a more generally aplicable criterion, that the sequences (conn) and (cron) are not P-recursive. On the other hand, we show that the residues of conn and cron modulo any xed power of 2 can be determined P-recursively. We consider also numbers scon of symmetric connected matchings. Unfortunately, their OGF satisses a more complicated diierential equation which we cannot handle. R esum e. C'est le r esum e frann cais. Matchings on the vertex set 2n] = f1; 2; : : : ; 2ng consist of n mutually disjoint two-element edges. One easily nds that their number mat n equals (2n ? 1)!! = 1 3 5 : : : (2n ? 1). Another classical result tells us that the number ncr n of noncrossing matchings on 2n] (no two edges fa; bg and fc; dg satisfy a < c < b < d) is the nth Catalan number: ncr n = 1 n+1 ? 2n n. How many matchings are there if their crossings are restricted in a more complicated way? In the present article we investigate numbers of such matchings, namely the numbers con n of connected matchings in which each two edges can be connected by a chain of consecutivly crossing edges, the numbers sco n of symmetric connected matchings which, in addition, are mirror symmetric, and the numbers cro n of crossing matchings in which each edge crosses another edge. We concentrate only on P-recursiveness of these numbers. Also, we touch upon some congruence properties. The sequences (mat n) and (ncr n) are trivially P-recursive but, as we prove, the sequences (con n) and (cro n) are not. First we remind the deenition of P-recursiveness and D-niteness. Then we introduce D A-niteness and review some facts on power series. In Theorem 1 we prove that if a sequence of numbers has OGF (ordinary generating function) that has zero convergence radius and satisses a certain diierential equation, then the sequence is far from being P-recursive. In Theorem 2 we apply this criterion to the sequences (con n) and (cro n). In Theorem 3 a more complicated …