A361592 Triangular array read by rows. T(n,k) is the number of labeled digraphs on [n] with exactly k strongly connected components of size 1, n>=0, 0<=k<=n.

1, 0, 1, 1, 0, 3, 18, 21, 0, 25, 1699, 1080, 774, 0, 543, 587940, 267665, 103860, 59830, 0, 29281, 750744901, 225144360, 64169325, 19791000, 10110735, 0, 3781503, 3556390155318, 672637205149, 126726655860, 29445913175, 7939815030, 3767987307, 0, 1138779265

Offset: 0

Author

Geoffrey Critzer, Mar 16 2023

Keywords

• nonn
• tabl

Examples

			Triangle begins:
       1;
       0,      1;
       1,      0,      3;
      18,     21,      0,    25;
    1699,   1080,    774,     0, 543;
  587940, 267665, 103860, 59830,   0, 29281;
  ...
		

Links

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

Crossrefs

Cf. A086366 (column k=0), A003024 (main diagonal), A053763 (row sums), A361590 (unlabeled version).

Programs

Mathematica

nn = 7; B[n_] := n! 2^Binomial[n, 2]; strong = Select[Import["https://oeis.org/A003030/b003030.txt", "Table"], Length@# == 2 &][[All, 2]];s[x_] := Total[strong Table[x^i/i!, {i, 1, 58}]]; ggfz[egfx_] := Normal[Series[egfx, {x, 0, nn}]] /.Table[x^i -> z^i/2^Binomial[i, 2], {i, 0, nn}];Table[Take[(Table[B[n], {n, 0, nn}] CoefficientList[Series[1/ggfz[Exp[-(s[x] - x + u x)]], {z, 0, nn}], {z,u}])[[i]], i], {i, 1, nn + 1}] // Grid