A322402 Triangle read by rows: The number of chord diagrams with n chords and k topologically connected components, 0 <= k <= n.

1, 0, 1, 0, 1, 2, 0, 4, 6, 5, 0, 27, 36, 28, 14, 0, 248, 310, 225, 120, 42, 0, 2830, 3396, 2332, 1210, 495, 132, 0, 38232, 44604, 29302, 14560, 6006, 2002, 429, 0, 593859, 678696, 430200, 204540, 81900, 28392, 8008, 1430, 0, 10401712, 11701926, 7204821, 3289296, 1263780, 431256, 129948, 31824, 4862

Offset

0,6

Comments

If all subsets are allowed instead of just pairs (chords), we get A324173. The rightmost column is A000108 (see Riordan). - Gus Wiseman, Feb 27 2019

Links

Table of n, a(n) for n=0..54.

P. Flajolet and M. Noy, Analytic Combinatorics of Chord Diagrams, in: Formal power series and algebraic combinatorics (FPSAC '00) Moscow, 2000, p 191-201, eq (2)

J. Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., 29 (1975), 215-222.

J. Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., 29 (1975), 215-222. [Annotated scanned copy]

Gus Wiseman, Chords diagrams with 3 chords, organized by number of components.

Formula

The g.f. satisfies g(z,w) = 1+w*A000699(w*g^2), where A000699(z) is the g.f. of A000699.

Example

From Gus Wiseman, Feb 27 2019: (Start)
Triangle begins:
  1
  0      1
  0      1      2
  0      4      6      5
  0     27     36     28     14
  0    248    310    225    120     42
  0   2830   3396   2332   1210    495    132
  0  38232  44604  29302  14560   6006   2002    429
  0 593859 678696 430200 204540  81900  28392   8008   1430
Row n = 3 counts the following chord diagrams (see link for pictures):
  {{1,3},{2,5},{4,6}}  {{1,2},{3,5},{4,6}}  {{1,2},{3,4},{5,6}}
  {{1,4},{2,5},{3,6}}  {{1,3},{2,4},{5,6}}  {{1,2},{3,6},{4,5}}
  {{1,4},{2,6},{3,5}}  {{1,3},{2,6},{4,5}}  {{1,4},{2,3},{5,6}}
  {{1,5},{2,4},{3,6}}  {{1,5},{2,3},{4,6}}  {{1,6},{2,3},{4,5}}
                       {{1,5},{2,6},{3,4}}  {{1,6},{2,5},{3,4}}
                       {{1,6},{2,4},{3,5}}
(End)

Crossrefs

Cf. A000699 (k = 1 column), A001147 (row sums), A000108 (diagonal), A002694 (subdiagonal k = n - 1).

Cf. A000096, A003436, A016098, A099947, A136653, A278990, A293157, A324173, A324323, A324327, A324328.

Sequence in context: A133144 A342384 A192134 * A196877 A098123 A066659

Adjacent sequences: A322399 A322400 A322401 * A322403 A322404 A322405

Keyword

nonn,tabl

Author

R. J. Mathar, Dec 06 2018

Extensions

Offset changed to 0 by Gus Wiseman, Feb 27 2019

Status

approved