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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.

%I A322402 #15 Mar 01 2019 08:05:57
%S A322402 1,0,1,0,1,2,0,4,6,5,0,27,36,28,14,0,248,310,225,120,42,0,2830,3396,
%T A322402 2332,1210,495,132,0,38232,44604,29302,14560,6006,2002,429,0,593859,
%U A322402 678696,430200,204540,81900,28392,8008,1430,0,10401712,11701926,7204821,3289296,1263780,431256,129948,31824,4862
%N A322402 Triangle read by rows: The number of chord diagrams with n chords and k topologically connected components, 0 <= k <= n.
%C A322402 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
%H A322402 P. Flajolet and M. Noy, <a href="http://algo.inria.fr/flajolet/Publications/FlNo00.pdf">Analytic Combinatorics of Chord Diagrams</a>, in: Formal power series and algebraic combinatorics (FPSAC '00) Moscow, 2000, <a href="https://doi.org/10.1007/978-3-662-04166-6">p 191-201</a>, eq (2)
%H A322402 J. Riordan, <a href="https://doi.org/10.1090/S0025-5718-1975-0366686-9">The distribution of crossings of chords joining pairs of 2n points on a circle</a>, Math. Comp., 29 (1975), 215-222.
%H A322402 J. Riordan, <a href="/A003480/a003480.pdf">The distribution of crossings of chords joining pairs of 2n points on a circle</a>, Math. Comp., 29 (1975), 215-222. [Annotated scanned copy]
%H A322402 Gus Wiseman, <a href="/A322402/a322402.png">Chords diagrams with 3 chords, organized by number of components</a>.
%F A322402 The g.f. satisfies g(z,w) = 1+w*A000699(w*g^2), where A000699(z) is the g.f. of A000699.
%e A322402 From _Gus Wiseman_, Feb 27 2019: (Start)
%e A322402 Triangle begins:
%e A322402   1
%e A322402   0      1
%e A322402   0      1      2
%e A322402   0      4      6      5
%e A322402   0     27     36     28     14
%e A322402   0    248    310    225    120     42
%e A322402   0   2830   3396   2332   1210    495    132
%e A322402   0  38232  44604  29302  14560   6006   2002    429
%e A322402   0 593859 678696 430200 204540  81900  28392   8008   1430
%e A322402 Row n = 3 counts the following chord diagrams (see link for pictures):
%e A322402   {{1,3},{2,5},{4,6}}  {{1,2},{3,5},{4,6}}  {{1,2},{3,4},{5,6}}
%e A322402   {{1,4},{2,5},{3,6}}  {{1,3},{2,4},{5,6}}  {{1,2},{3,6},{4,5}}
%e A322402   {{1,4},{2,6},{3,5}}  {{1,3},{2,6},{4,5}}  {{1,4},{2,3},{5,6}}
%e A322402   {{1,5},{2,4},{3,6}}  {{1,5},{2,3},{4,6}}  {{1,6},{2,3},{4,5}}
%e A322402                        {{1,5},{2,6},{3,4}}  {{1,6},{2,5},{3,4}}
%e A322402                        {{1,6},{2,4},{3,5}}
%e A322402 (End)
%Y A322402 Cf. A000699 (k = 1 column), A001147 (row sums), A000108 (diagonal), A002694 (subdiagonal k = n - 1).
%Y A322402 Cf. A000096, A003436, A016098, A099947, A136653, A278990, A293157, A324173, A324323, A324327, A324328.
%K A322402 nonn,tabl
%O A322402 0,6
%A A322402 _R. J. Mathar_, Dec 06 2018
%E A322402 Offset changed to 0 by _Gus Wiseman_, Feb 27 2019