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A362226 Triangular array read by rows. T(n,k) is the number of labeled digraphs on [n] with exactly k isolated strongly connected components, n>=0, 0<=k<=n.

%I A362226 #11 Apr 12 2023 11:31:53
%S A362226 1,0,1,2,1,1,36,24,3,1,2240,1762,87,6,1,462720,577000,8630,215,10,1,
%T A362226 332613632,737645836,3455820,26085,435,15,1,867410804736,
%U A362226 3525456796232,5166693532,12154030,61775,777,21,1,8503156728135680,63526200994115056,28215577119548,20705805988,32624585,125776,1274,28,1
%N A362226 Triangular array read by rows.  T(n,k) is the number of labeled digraphs on [n] with exactly k isolated strongly connected components, n>=0, 0<=k<=n.
%C A362226 Here, a strongly connected component is isolated if it is both an in-component and an out-component.  A component is an in-component (out-component) if it corresponds to a node with outdegree (indegree) zero in the condensation of the digraph.
%H A362226 E. de Panafieu and S. Dovgal, <a href="https://arxiv.org/abs/1903.09454">Symbolic method and directed graph enumeration</a>, arXiv:1903.09454 [math.CO], 2019.
%H A362226 R. W. Robinson, <a href="http://cobweb.cs.uga.edu/~rwr/publications/components.pdf">Counting digraphs with restrictions on the strong components</a>, Combinatorics and Graph Theory '95 (T.-H. Ku, ed.), World Scientific, Singapore (1995), 343-354.
%F A362226 E.g.f.:  exp((u-1)*S(z))*D(z) where S(z) is the e.g.f. for A003030 and D(z) is the e.g.f. for A053763.
%e A362226        1;
%e A362226        0,      1;
%e A362226        2,      1,    1;
%e A362226       36,     24,    3,   1;
%e A362226     2240,   1762,   87,   6,  1;
%e A362226   462720, 577000, 8630, 215, 10, 1;
%e A362226  ...
%t A362226 nn = 8; strong = Select[Import["https://oeis.org/A003030/b003030.txt", "Table"],
%t A362226    Length@# == 2 &][[All, 2]]; s[z_] := Total[strong Table[z^i/i!, {i, 1, 58}]];
%t A362226 d[z_] := Sum[2^(n (n - 1)) z^n/n!, {n, 0, nn}]; Table[Take[(Table[n!, {n, 0, nn}] CoefficientList[ Series[Exp[(u - 1) s[z]] d[z], {z, 0, nn}], {z, u}])[[i]],
%t A362226    i], {i, 1, nn + 1}] // Grid
%Y A362226 Cf. A217580, A361579, A003030, A053763.
%K A362226 nonn,tabl
%O A362226 0,4
%A A362226 _Geoffrey Critzer_, Apr 11 2023