A054733
Triangle of number of (weakly) connected unlabeled digraphs with n nodes and k arcs (n >=2, k >= 1).
Original entry on oeis.org
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, 1127, 1477, 1665, 1489, 1154, 707, 379, 154, 61, 16, 5, 1, 1, 0, 0, 0, 0, 91, 582, 2432, 7694, 19646, 42148, 77305, 122953, 170315, 206982, 220768, 207301, 171008
Offset: 2
1,1;
0,3,4,4,1,1;
0,0,8,22,37,47,38,27,13,5,1,1;
the last batch giving the numbers of connected digraphs with 4 nodes and from 1 to 12 arcs.
- F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973.
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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)}
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}
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))}
G(n, x)={my(s=0); forpart(p=n, s+=permcount(p)*edges(p, i->1+x^i)); s/n!}
row(n)={Vecrev(polcoef(InvEulerMTS(sum(i=0, n, G(i, y)*x^i, O(x*x^n))), n)/y)}
{ for(n=2, 6, print(row(n))) } \\ Andrew Howroyd, Jan 28 2022
A053418
Number of unlabeled directed graphs with n arcs and no isolated vertices.
Original entry on oeis.org
1, 1, 5, 17, 80, 365, 1981, 11222, 69511, 455663, 3169244, 23170347, 177513359, 1418920570, 11798710013, 101778754655, 908722427531, 8380602471646, 79692654473866, 780142956502644, 7851084073063731, 81120767066417308
Offset: 0
A350789
Triangle read by rows: T(n,k) is the number of unlabeled weakly connected digraphs with n arcs and k vertices, k = 1..n+1.
Original entry on oeis.org
1, 0, 1, 0, 1, 3, 0, 0, 4, 8, 0, 0, 4, 22, 27, 0, 0, 1, 37, 108, 91, 0, 0, 1, 47, 326, 582, 350, 0, 0, 0, 38, 667, 2432, 3024, 1376, 0, 0, 0, 27, 1127, 7694, 17314, 16008, 5743, 0, 0, 0, 13, 1477, 19646, 74676, 117312, 84494, 24635
Offset: 0
Triangle begins:
1;
0, 1;
0, 1, 3;
0, 0, 4, 8;
0, 0, 4, 22, 27;
0, 0, 1, 37, 108, 91;
0, 0, 1, 47, 326, 582, 350;
0, 0, 0, 38, 667, 2432, 3024, 1376;
0, 0, 0, 27, 1127, 7694, 17314, 16008, 5743;
...
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\\ See A054733 for G, InvEulerMTS
T(n)={my(p=InvEulerMTS(sum(i=0, n, G(i, y+O(y^n))*x^i, O(x*x^n)))); vector(n, n, Vec(O(x^n)+polcoef(p,n-1,y)/x, -n))}
{ my(A=T(10)); for(n=1, #A, print(A[n])) }
A350915
Number of weakly connected oriented graphs with n arcs.
Original entry on oeis.org
1, 1, 3, 10, 39, 169, 876, 4834, 29316, 189054, 1294382, 9321232, 70326820, 553433559, 4528840412, 38432156859, 337454775045, 3059843449398, 28602687303185, 275222034228537, 2722343346822614, 27647618196693537, 287970349621911635, 3073082817450997700, 33568654163238906968
Offset: 0
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\\ See A350734 for G, InvEulerMTS.
seq(n)=Vec(subst(Pol(InvEulerMTS(sum(i=0, n, G(i, y+O(y^n))*x^i, O(x*x^n)))), x, 1))
Showing 1-4 of 4 results.