A081054 Crossing matchings: linear chord diagrams with 2n nodes and n arcs in which each arc crosses another arc.
1, 0, 1, 4, 31, 288, 3272, 43580, 666143, 11491696, 220875237, 4681264432, 108475235444, 2728591657920, 74051386322580, 2156865088819692, 67113404608820943, 2221948578439255200, 77990056655776149179
Offset: 0
Examples
The 4 crossing matchings on nodes 1, 2, ..., 6 are {13, 25, 46}, {14, 25, 36}, {15, 24, 36} and {14, 26, 35}.
Links
- Olivia Beckwith, Victor Luo, Stephen J. Miller, Karen Shen, Nicholas Triantafillou, Distribution of Eigenvalues of Weighted, Structured Matrix Ensembles, arXiv:1112.3719 [math.PR], 2011-2012.
- Olivia Beckwith, Victor Luo, Stephen J. Miller, Karen Shen, Nicholas Triantafillou, Distribution of Eigenvalues of Weighted, Structured Matrix Ensembles, Electronic Journal of Combinatorial Number Theory, Volume 15 (2015) #A21.
- M. Klazar, Non-P-recursiveness of numbers of matchings or linear chord diagrams with many crossings, Advances in Appl. Math., Vol. 30 (2003), pp. 126-136.
- Alexander Stoimenow, On enumeration of chord diagrams and asymptotics of Vassiliev invariants, Dissertation, Mathematik und Informatik, University of Berlin, 1998; see chapter 3.
Programs
-
Mathematica
a[n_] := a[n]=Module[{x, y, z, i}, y=Sum[a[i]x^i, {i, 0, n-1}]+z*x^n+O[x]^(n+1); Solve[D[y, x]==(-1+y-x^2y^3)/(2x^2y(1+x*y)), z][[1, 1, 2]]]
Formula
The g.f. (a formal power series) F = 1 + x^2 + 4*x^3 + ... satisfies the differential equation F' = (-x^2*F^3 + F - 1)/(2*x^3*F^2 + 2*x^2*F).
a(n) is asymptotic to (2n)!/(e 2^n n!). In other words, the probability that a random matching is a crossing matching is asymptotic to 1/e; see Lemma 3.12 of Stoimenow reference. - Benoit Cloitre, Apr 18 2003; corrected by Dean Hickerson, Apr 21 2003