A057271 - OEIS

%I #16 Jan 16 2022 17:46:36

%S 1,0,2,1,0,0,6,20,15,6,1,0,0,0,24,234,672,908,792,495,220,66,12,1,0,0,

%T 0,0,120,2544,16880,55000,111225,161660,183006,167660,125945,77520,

%U 38760,15504,4845,1140,190,20,1

%N Triangle T(n,k) of number of digraphs with a source and a sink on n labeled nodes and k arcs, k=0,1,..,n*(n-1).

%D V. Jovovic, G. Kilibarda, Enumeration of labeled initially-finally connected digraphs, Scientific review, Serbian Scientific Society, 19-20 (1996), p. 245.

%H Andrew Howroyd, <a href="/A057271/b057271.txt">Table of n, a(n) for n = 1..2680</a> (rows 1..20)

%H 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.

%H 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.

%e Triangle starts:

%e [1] 1;

%e [2] 0,2,1;

%e [3] 0,0,6,20,15,6,1;

%e [4] 0,0,0,24,234,672,908,792,495,220,66,12,1;

%e   ...

%e The number of digraphs with a source and a sink on 3 labeled nodes is 48 = 6+20+15+6+1.

%o (PARI) \\ Following Eqn 20 in the Robinson reference.

%o Z(p,f)={my(n=serprec(p,x)); serconvol(p, sum(k=0, n-1, x^k*f(k), O(x^n)))}

%o G(e,p)={Z(p, k->1/e^(k*(k-1)/2))}

%o U(e,p)={Z(p, k->e^(k*(k-1)/2))}

%o DigraphEgf(n,e)={sum(k=0, n, e^(k*(k-1))*x^k/k!, O(x*x^n) )}

%o StrongD(n,e=2)={-log(U(e, 1/G(e, DigraphEgf(n, e))))}

%o 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))) ))}

%o row(n)={Vecrev(InitFinally(n, 1+'y)[n]) }

%o { for(n=1, 5, print(row(n))) } \\ _Andrew Howroyd_, Jan 16 2022

%Y Row sums give A049524.

%Y The unlabeled version is A057278.

%Y Cf. A057272, A057273, A057274, A057275, A062735, A350791.

%K nonn,tabf

%O 1,3

%A _Vladeta Jovovic_, Goran Kilibarda, Sep 14 2000