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A322402
Triangle read by rows: The number of chord diagrams with n chords and k topologically connected components, 0 <= k <= n.
5
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
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. [Annotated scanned copy]
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).
Sequence in context: A133144 A342384 A192134 * A196877 A098123 A066659
KEYWORD
nonn,tabl
AUTHOR
R. J. Mathar, Dec 06 2018
EXTENSIONS
Offset changed to 0 by Gus Wiseman, Feb 27 2019
STATUS
approved