A350488 -id:A350488 - cp's OEIS frontend

A350415 Number of acyclic digraphs on n unlabeled nodes with a global source (or sink).

1, 1, 3, 16, 164, 3341, 138101, 11578037, 1961162564, 668678055847, 457751797355605, 628137837068751147, 1726130748679532455689, 9493834992383031007906911, 104476428350838383854529661007, 2299979227717819421763629684068904

Offset: 1

Andrew Howroyd, Dec 29 2021

nonn

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.

Andrew Howroyd, Table of n, a(n) for n = 1..50
Marcel et al., Is there a formula for the number of st-dags (DAG with 1 source and 1 sink) with n vertices?, MathOverflow, 2021.

The labeled case is A003025.
Row sums of A350488.
A diagonal of A122078.
Cf. A003087, A345258, A350360.

A350415seq(16) \\ See PARI link in A122078 for program code.

A350491 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.

Original entry on oeis.org

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

Offset: 1

Andrew Howroyd, Jan 08 2022

			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;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1350 (rows 1..20)

Row sums are A345258.
Column sums are A350492.
Cf. A122078, A350447, A350449, A350488.

\\ See PARI link in A122078 for program code.
{ my(A=A350491rows(7)); for(i=1, #A, print(A[i])) }

A350487 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.

Original entry on oeis.org

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

Offset: 1

Andrew Howroyd, Jan 01 2022

			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;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1350 (rows 1..20)
Marcel et al., Is there a formula for the number of st-dags (DAG with 1 source and 1 sink) with n vertices?, MathOverflow, 2021.

Row sums are A003025.
Leading diagonal is A000169.
The unlabeled version is A350488.
Cf. A081064.

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])) }

A350490 Number of unlabeled acyclic digraphs with n arcs and a global source.

Original entry on oeis.org

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

Offset: 0

Andrew Howroyd, Jan 08 2022

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..40

Column sums of A350488.

Cf. A122078, A350415, A350451, A350492.

A350490seq(20) \\ See PARI link in A122078 for program code.

cp's OEIS Frontend

A350415 Number of acyclic digraphs on n unlabeled nodes with a global source (or sink).

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A350491 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.

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PARI

A350487 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.

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PARI

A350490 Number of unlabeled acyclic digraphs with n arcs and a global source.

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Author

Keywords

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Programs

PARI