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Number of acyclic digraphs on n unlabeled nodes with a global source (or sink).
+10
10
1, 1, 3, 16, 164, 3341, 138101, 11578037, 1961162564, 668678055847, 457751797355605, 628137837068751147, 1726130748679532455689, 9493834992383031007906911, 104476428350838383854529661007, 2299979227717819421763629684068904
COMMENTS
A local source (also called an out-node) is a node whose in-degree is zero. In the case of an acyclic digraph with only one local source, the source is also a global source.
PROG
(PARI) A350415seq(16) \\ See PARI link in A122078 for program code.
Triangle read by rows: T(n,k) is the number of acyclic digraphs on n unlabeled nodes with k arcs and a global source and sink, n >= 1, k = 0..n*(n-1)/2.
+10
4
1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 4, 4, 1, 0, 0, 0, 0, 1, 9, 25, 32, 22, 8, 1, 0, 0, 0, 0, 0, 1, 17, 92, 259, 441, 496, 379, 195, 66, 13, 1, 0, 0, 0, 0, 0, 0, 1, 28, 259, 1286, 4026, 8754, 13930, 16686, 15289, 10785, 5842, 2397, 722, 151, 19, 1
EXAMPLE
Triangle begins:
[1] 1;
[2] 0, 1;
[3] 0, 0, 1, 1;
[4] 0, 0, 0, 1, 4, 4, 1;
[5] 0, 0, 0, 0, 1, 9, 25, 32, 22, 8, 1;
[6] 0, 0, 0, 0, 0, 1, 17, 92, 259, 441, 496, 379, 195, 66, 13, 1;
...
PROG
(PARI) \\ See PARI link in A122078 for program code. { my(A=A350491rows(7)); for(i=1, #A, print(A[i])) }
Triangle read by rows: T(n,k) is the number of acyclic digraphs on n labeled nodes with k arcs and a global source, n >= 1, k = 0..n*(n-1)/2.
+10
3
1, 0, 2, 0, 0, 9, 6, 0, 0, 0, 64, 132, 96, 24, 0, 0, 0, 0, 625, 2640, 4850, 4900, 2850, 900, 120, 0, 0, 0, 0, 0, 7776, 55800, 186480, 379170, 516660, 491040, 328680, 152640, 46980, 8640, 720, 0, 0, 0, 0, 0, 0, 117649, 1286670, 6756120, 22466010
EXAMPLE
Triangle begins:
[1] 1;
[2] 0, 2;
[3] 0, 0, 9, 6;
[4] 0, 0, 0, 64, 132, 96, 24;
[5] 0, 0, 0, 0, 625, 2640, 4850, 4900, 2850, 900, 120;
...
PROG
(PARI)
T(n)={my(a=vector(n)); a[1]=1; for(n=2, #a, a[n]=sum(k=1, n-1, (-1)^(k-1)*binomial(n, k)*((1+'y)^(n-k)-1)^k*a[n-k])); [Vecrev(p) | p <- a]}
{ my(A=T(6)); for(n=1, #A, print(A[n])) }
Number of unlabeled acyclic digraphs with n arcs and a global source.
+10
3
1, 1, 2, 5, 15, 50, 191, 795, 3625, 17781, 93319, 519732, 3056185, 18880345, 122066437, 823118881, 5772473719, 41995743085, 316258151517, 2460542117337, 19743544526463, 163138014893817, 1386173917051704, 12096681529032989, 108293699332498981
PROG
(PARI) A350490seq(20) \\ See PARI link in A122078 for program code.
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