|
|
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).
|
|
8
|
|
|
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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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
|
|
|
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}
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|