Non-p-recursiveness of Numbers of Matchings or Linear Chord Diagrams with Many Crossings

The number conn 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, cron 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 (conn) and (cron) are not P-recursive. On the other hand, we show that the residues of conn and cron modulo any fixed power of 2 can be determined P-recursively. We consider also the numbers scon of symmetric connected matchings. Unfortunately, their generating function satisfies a complicated differential equation which we cannot handle. Matchings on the vertex set [2n] = {1, 2, . . . , 2n} consist of n mutually disjoint two-element edges. One finds easily that their number matn equals (2n− 1)!! = 1 · 3 · 5 · . . . · (2n− 1). Another classical result tells us that the number ncrn of noncrossing matchings on [2n] (no two edges {a, b} and {c, d} satisfy a < c < b < d) is the nth Catalan number: ncrn = 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 conn of connected matchings in which each two edges can be connected by a chain of consecutivly crossing edges, the numbers scon of symmetric connected matchings which, in addition, are mirror symmetric, and the numbers cron of crossing matchings in which each edge crosses another edge. We concentrate only on P-recursiveness of these numbers. Also, we touch upon some modular properties. The sequences (matn) and (ncrn) are trivially P-recursive but, as we prove, the sequences (conn) and (cron) are not. First we remind the definition of P-recursiveness and D-finiteness. Then we introduce DAfiniteness and review some facts on power series. In Theorem 1 we prove that if a sequence of numbers has an OGF (ordinary generating function) that has zero convergence radius and satisfies a certain differential equation, then the sequence is far from being P-recursive. In Theorem 2 we apply this criterion to the sequences (conn) and (cron). In Theorem 3 a more complicated differential equation is derived for the OGF of the sequence (scon). Finally, in Theorem 4 we show that modulo 2 the sequences (conn) and (cron) coincide with certain P-recursive, in fact algebraic, sequences. Symbols Z and N denote the sets of integers {. . . ,−1, 0, 1, . . .} and {1, 2, . . .}. C denotes the set of complex numbers. A sequence of complex numbers (an)n≥0 is called P-recursive if there exist polynomials P0, P1, . . . , Pj ∈ C[x], P0 6= 0, such that P0(n)an + P1(n)an−1 + · · ·+ Pj(n)an−j = 0 holds for each integer n, n ≥ j. Many combinatorial counting sequences are P-recursive, for instance (matn), (ncrn), Schröder numbers, and numbers of labelled k-regular graphs (Gessel [7]). But some are not, for instance Bell numbers and numbers of integer partitions. We write C[[x1, . . . , xk]] for the ring of power series with complex coefficients and variables x1, . . . , xk. A power series F ∈ C[[x]] is D-finite if F solves the linear differential equation R0F (m) +R1F (m−1) + · · ·+Rm−1F ′ +RmF +Rm+1 = 0 Supported by project LN00A056 of The Ministery of Education of the Czech Republic.