A350790 Number of digraphs on n labeled nodes with a global source and sink.

1, 2, 12, 432, 64240, 33904800, 61721081184, 394586260943616, 9146766152111641344, 792073976107698469670400, 261895415169919230764987845120, 335402460348866803020064114666616832, 1678893205649791601327398844631544110815232

Offset: 1

Author

Andrew Howroyd, Jan 16 2022

Keywords

• nonn

Comments

This sequence differs from A049524 in that the source and sink are restricted to being single nodes.

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..50
• R. W. Robinson, Counting digraphs with restrictions on the strong components, Combinatorics and Graph Theory '95 (T.-H. Ku, ed.), World Scientific, Singapore (1995), 343-354.

Crossrefs

The unlabeled version is A350794.

Row sums of A350791.

Cf. A049524, A165950, A350792, A003030.

Programs

Mathematica

nn = 15; strong = Select[Import["https://oeis.org/A003030/b003030.txt", "Table"],
   Length@# == 2 &][[All, 2]]; s[z_] := Total[strong Table[z^i/i!, {i, 1, 58}]];
ggf[egf_] := Normal[Series[egf, {z, 0, nn}]] /. Table[z^i -> z^i/2^Binomial[i, 2], {i, 1, nn + 1}];egf[ggf_] := Normal[Series[ggf, {z, 0, nn}]] /.Table[z^i -> z^i*2^Binomial[i, 2], {i, 1, nn + 1}];Table[n!, {n, 0, nn}] CoefficientList[
Series[z - z^2 + Exp[(u - 1) (v - 1) s[ z]] egf[ggf[z Exp[(u - 1) s[z]]]*1/ggf[Exp[-s[z]]]*ggf[z Exp[(v - 1) s[ z]]]] /. {u -> 0, v -> 0}, {z, 0, nn}], z] (* Geoffrey Critzer, Apr 17 2023 *)

PARI

InitFinallyV(12) \\ See A350791 for program code.

Formula

For n >= 3, a(n) = 2*n*(n-1)*A003030(n-1) (Robinson equation 22). - Geoffrey Critzer, Apr 17 2023