A165950 Number of acyclic digraphs on n labeled nodes with one source and one sink.

1, 2, 12, 216, 10600, 1306620, 384471444, 261548825328, 402632012394000, 1381332938730123060, 10440873023366019273820, 172308823347127690038311496, 6163501139185639837183141411320, 474942255590583211554917995123517868, 78430816994991932467786587093292327531620

Offset: 1

Author

Vladeta Jovovic, Oct 01 2009

Keywords

• nonn

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..50
• Antoine Genitrini, Martin Pépin, and Alfredo Viola, Unlabelled ordered DAGs and labelled DAGs: constructive enumeration and uniform random sampling, hal-03029381 [math.CO], [cs.DM], [cs.DS], 2020.
• Ira Gessel, Counting Acyclic Digraphs by Sources and Sinks
• 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.

Crossrefs

The unlabeled version is A345258.

Cf. A003024, A003025, A049524, A361210.

Programs

Mathematica

nn = 10; B[n_] := n! 2^Binomial[n, 2];e[z_] := Sum[z^n/B[n], {n, 0, nn}];
egf[ggf_] := Normal[Series[ggf, {z, 0, nn}]] /. Table[z^i -> z^i*2^Binomial[i, 2], {i, 1, nn + 1}]; Map[ Coefficient[#, u v] &,Table[n!, {n, 0, nn}] CoefficientList[ Series[Exp[(u - 1) (v - 1) z] egf[e[(u - 1) z]*1/e[-z]*e[(v - 1) z]], {z, 0, nn}], z]] (* Geoffrey Critzer, Apr 15 2023 *)

PARI

\\ see Marcel et al. link. B(n) is A003025 as vector.
B(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)*(2^(n-k)-1)^k*a[n-k])); a}
seq(n)={my(a=vector(n), b=B(n)); a[1]=1; for(n=2, #a, a[n]=sum(k=1, n-1, (-1)^(k-1) * binomial(n,k) * k * (2^(n-k)-1)^k * b[n-k])); a} \\ Andrew Howroyd, Jan 01 2022

Extensions

a(1)=1 inserted and terms a(13) and beyond from Andrew Howroyd, Jan 01 2022