cp's OEIS Frontend

A054733 Triangle of number of (weakly) connected unlabeled digraphs with n nodes and k arcs (n >=2, k >= 1).

%I A054733 #22 Feb 19 2022 20:25:52
%S A054733 1,1,0,3,4,4,1,1,0,0,8,22,37,47,38,27,13,5,1,1,0,0,0,27,108,326,667,
%T A054733 1127,1477,1665,1489,1154,707,379,154,61,16,5,1,1,0,0,0,0,91,582,2432,
%U A054733 7694,19646,42148,77305,122953,170315,206982,220768,207301,171008
%N A054733 Triangle of number of (weakly) connected unlabeled digraphs with n nodes and k arcs (n >=2, k >= 1).
%D A054733 F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973.
%H A054733 Andrew Howroyd, <a href="/A054733/b054733.txt">Table of n, a(n) for n = 2..2661</a> (rows 2..20)
%H A054733 R. J. Mathar, <a href="http://arxiv.org/abs/1709.09000">Statistics on Small Graphs</a>, arXiv:1709.09000  (2017) Table 75.
%e A054733 1,1;
%e A054733 0,3,4,4,1,1;
%e A054733 0,0,8,22,37,47,38,27,13,5,1,1;
%e A054733 the last batch giving the numbers of connected digraphs with 4 nodes and from 1 to 12 arcs.
%o A054733 (PARI)
%o A054733 InvEulerMTS(p)={my(n=serprec(p,x)-1, q=log(p), vars=variables(p)); sum(i=1, n, moebius(i)*substvec(q + O(x*x^(n\i)), vars, apply(v->v^i,vars))/i)}
%o A054733 permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
%o A054733 edges(v, t) = {prod(i=2, #v, prod(j=1, i-1, my(g=gcd(v[i], v[j])); t(v[i]*v[j]/g)^(2*g) )) * prod(i=1, #v, my(c=v[i]); t(c)^(c-1))}
%o A054733 G(n, x)={my(s=0); forpart(p=n, s+=permcount(p)*edges(p, i->1+x^i)); s/n!}
%o A054733 row(n)={Vecrev(polcoef(InvEulerMTS(sum(i=0, n, G(i, y)*x^i, O(x*x^n))), n)/y)}
%o A054733 { for(n=2, 6, print(row(n))) } \\ _Andrew Howroyd_, Jan 28 2022
%Y A054733 Cf. A000238 (leading diagonal), A003085 (row sums), A053454 (column sums), A062735 (labeled).
%Y A054733 Cf. A052283 (not necessarily connected), A283753 (another version), A057276 (strongly connected), A350789 (transpose).
%K A054733 easy,nonn,tabf
%O A054733 2,4
%A A054733 _Vladeta Jovovic_, Apr 21 2000