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Andrew Howroyd, <a href="/A057271/b057271.txt">Table of n, a(n) for n = 1..2680</a> (rows 1..20)
Andrew Howroyd, <a href="/A057271/b057271.txt">Table of n, a(n) for n = 1..2680</a>
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.
[1], 1;
[2] 0,2,1],;
[3] 0,0,6,20,15,6,1],;
[4] 0,0,0,24,234,672,908,792,495,220,66,12,1],;
...
...
Number The number of digraphs with a source and a sink on 3 labeled nodes is 48 = 6+20+15+6+1.
(PARI) \\ Following Eqn 20 in Robinson reference.
Z(p, f)={my(n=serprec(p, x)); serconvol(p, sum(k=0, n-1, x^k*f(k), O(x^n)))}
G(e, p)={Z(p, k->1/e^(k*(k-1)/2))}
U(e, p)={Z(p, k->e^(k*(k-1)/2))}
DigraphEgf(n, e)={sum(k=0, n, e^(k*(k-1))*x^k/k!, O(x*x^n) )}
StrongD(n, e=2)={-log(U(e, 1/G(e, DigraphEgf(n, e))))}
InitFinally(n, e=2)={my(S=StrongD(n, e)); Vec(serlaplace( S - S^2 + exp(S) * U(e, G(e, S*exp(-S))^2*G(e, DigraphEgf(n, e))) ))}
row(n)={Vecrev(InitFinally(n, 1+'y)[n]) }
{ for(n=1, 5, print(row(n))) } \\ Andrew Howroyd, Jan 16 2022
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V. Jovovic, G. Kilibarda, Enumeration of labeled quasi-initially connected digraphs, Discrete Math., 224 (2000),151-163.
V. Jovovic and G. Kilibarda, <a href="http://dx.doi.org/10.1016/S0012-365X(00)00112-6">Enumeration of labeled quasi-initially connected digraphs</a>, Discrete Math., 224 (2000), 151-163.
Triangle starts:
[1],
[0,2,1],
[0,0,6,20,15,6,1],
[0,0,0,24,234,672,908,792,495,220,66,12,1],
...
[1],[0,2,1],[0,0,6,20,15,6,1],[0,0,0,24,234,672,908,792,495,220,66,12,1],...; Number of digraphs with a source and a sink on 3 labeled nodes is 48=6+20+15+6+1.
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