A058876 Triangle read by rows: T(n,k) = number of labeled acyclic digraphs with n nodes, containing exactly n+1-k points of in-degree zero (n >= 1, 1<=k<=n).

1, 1, 2, 1, 9, 15, 1, 28, 198, 316, 1, 75, 1610, 10710, 16885, 1, 186, 10575, 211820, 1384335, 2174586, 1, 441, 61845, 3268125, 64144675, 416990763, 654313415, 1, 1016, 336924, 43832264, 2266772550, 44218682312, 286992935964, 450179768312

Offset

1,3

References

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 19, (1.6.4).

R. W. Robinson, Counting labeled acyclic digraphs, pp. 239-273 of F. Harary, editor, New Directions in the Theory of Graphs. Academic Press, NY, 1973.

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

R. W. Robinson, Enumeration of acyclic digraphs, Manuscript. (Annotated scanned copy)

Formula

Harary and Prins (following Robinson) give a recurrence.

Example

Triangle begins:
  1;
  1,  2;
  1,  9,   15;
  1, 28,  198,   316;
  1, 75, 1610, 10710, 16885;
  ...

Mathematica

a[p_, k_] :=a[p, k] =If[p == k, 1, Sum[Binomial[p, k]*a[p - k, n]*(2^k - 1)^n*2^(k (p - k - n)), {n, 1, p - k}]];
Map[Reverse, Table[Table[a[p, k], {k, 1, p}], {p, 1, 6}]] // Grid (* Geoffrey Critzer, Aug 29 2016 *)

Prog

(PARI)
A058876(n)={my(v=vector(n)); for(n=1, #v, v[n]=vector(n, i, if(i==n, 1, my(u=v[n-i]); sum(j=1, #u, 2^(i*(#u-j))*(2^i-1)^j*binomial(n, i)*u[j])))); v}
{ my(T=A058876(10)); for(n=1, #T, print(Vecrev(T[n]))) } \\ Andrew Howroyd, Dec 27 2021

Crossrefs

Columns give A058877, A060337.

Diagonals give A003025, A003026, A060335.

Row sums give A003024.

Cf. A122078 (unlabeled case).

Sequence in context: A180001 A204371 A199887 * A214884 A083162 A178075

Adjacent sequences: A058873 A058874 A058875 * A058877 A058878 A058879

Keyword

nonn,easy,tabl

Author

N. J. A. Sloane, Jan 07 2001

Extensions

More terms from Vladeta Jovovic, Apr 10 2001

Status

approved