A362013 Triangular array read by rows. T(n,k) is the number of labeled directed graphs on [n] with exactly k strongly connected components of size 1 with outdegree zero, n>=0, 0<=k<=n.

1, 0, 1, 1, 2, 1, 27, 27, 9, 1, 2401, 1372, 294, 28, 1, 759375, 253125, 33750, 2250, 75, 1, 887503681, 171774906, 13852815, 595820, 14415, 186, 1, 3938980639167, 437664515463, 20841167403, 551353635, 8751645, 83349, 441, 1, 67675234241018881, 4263006881324024, 117484441611292, 1850148686792, 18210124870, 114709448, 451612, 1016, 1

Offset

0,5

Links

Table of n, a(n) for n=0..44.

E. de Panafieu and S. Dovgal, Symbolic method and directed graph enumeration, arXiv:1903.09454 [math.CO], 2019.

Example

Triangle T(n,k) begins:
       1;
       0,      1;
       1,      2,     1;
      27,     27,     9,    1;
    2401,   1372,   294,   28,  1;
  759375, 253125, 33750, 2250, 75, 1;
  ...

Mathematica

nn = 6; B[n_] := n! 2^Binomial[n, 2] ; 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, 0, nn}]; Table[ Take[(Table[B[n], {n, 0, nn}] CoefficientList[   Series[ggf[Exp[(u - 1) z]]/ggf[Exp[-s[z]]], {z, 0, nn}], {z, u}])[[i]], i], {i, 1, nn + 1}]

Crossrefs

Cf. A086206 (column k=0), A053763 (row sums), A361592, A350792 (a subclass of the digraphs for the case k=1 of this sequence), A003028.

Sequence in context: A235031 A138955 A089963 * A336911 A322230 A353910

Adjacent sequences: A362010 A362011 A362012 * A362014 A362015 A362016

Keyword

nonn,tabl

Author

Geoffrey Critzer, Apr 03 2023

Status

approved