A143024 Triangle read by rows: T(n,k) is the number of non-crossing connected graphs on n nodes on a circle having root (a distinguished node) of degree 1 and having k edges (n >= 2, 1 <= k <= 2n-4).
1, 0, 2, 0, 0, 7, 2, 0, 0, 0, 30, 20, 4, 0, 0, 0, 0, 143, 156, 65, 10, 0, 0, 0, 0, 0, 728, 1120, 720, 224, 28, 0, 0, 0, 0, 0, 0, 3876, 7752, 6783, 3192, 798, 84, 0, 0, 0, 0, 0, 0, 0, 21318, 52668, 58520, 36960, 13860, 2904, 264, 0, 0, 0, 0, 0, 0, 0, 0, 120175, 354200, 478170
Offset: 2
Examples
T(3,2)=2 because we have {AB,BC} and {AC, BC} (A is the root).
Triangle starts:
1;
0, 2;
0, 0, 7, 2;
0, 0, 0, 30, 20, 4;
0, 0, 0, 0, 143, 156, 65, 10;
Links
- C. Domb and A. J. Barrett, Enumeration of ladder graphs, Discrete Math. 9 (1974), 341-358.
- P. Flajolet and M. Noy, Analytic combinatorics of non-crossing configurations, Discrete Math., 204, 203-229, 1999.
Programs
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Maple
T:=proc(n,k) options operator, arrow: 2*binomial(k-2,n-3)*binomial(3*n-5,2*n-k-4)/(n-2) end proc: 1; for n from 3 to 10 do 0, seq(T(n,k),k=2..2*n-4) end do; % yields sequence in triangular form
Formula
T(n,k) = 2*binomial(k-2, n-3)*binomial(3n-5, 2n-k-4)/(n-2) (n >= 3, 2 <= k <= 2n-4); T(2,1)=1; T(2,k)=0 (k >= 2).
The trivariate g.f. G=G(t,s,z) for non-crossing connected graphs on nodes on a circle, with respect to number of nodes (marked by z), number of edges (marked by t) and degree of root (marked by s) is G=z + tszg^2/[z-ts(g - z + g^2)], where g=g(t,z) satisfies tg^3 + tg^2 - (1 + 2t)zg +(1 + t)z^2 = 0 (see Domb & Barrett, Eq. (47); Flajolet & Noy, Eq. (18)).
Comments