A303831
Birooted graphs: number of unlabeled connected graphs with n nodes rooted at 2 indistinguishable roots.
Original entry on oeis.org
0, 1, 3, 16, 98, 879, 11260, 230505, 7949596, 483572280, 53011686200, 10589943940654, 3880959679322754, 2623201177625659987, 3286005731275218388682, 7663042204550840483139108, 33407704152242477510352455230, 273327599183687887638526170380380
Offset: 1
A126122
Number of edge-rooted unlabeled graphs on n nodes.
Original entry on oeis.org
0, 1, 3, 14, 74, 572, 6564, 125120, 4147832, 247183688, 26814414976, 5327509126272, 1946813194004224, 1313879013920844480, 1644521424392726492800, 3833402238753872010743808, 16708198671847617602943683072, 136682601082422255664288717142080
Offset: 1
a(2)=1: the tree with 2 nodes and a rooted edge. a(3)=3: (i) the linear tree with one of the two edges rooted, (ii) the triangle graph with one of the three edges rooted, (iii) the disconnected graph with a single disconnected node and a tree with 2 nodes and a marked edge. - _R. J. Mathar_, May 01 2018
- F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973.
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permcount[v_] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length@v, i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];
edges[v_] := (Sum[GCD[v[[i]], v[[j]]], {i, 2, Length@v}, {j, 1, i - 1}] + Total[Quotient[v, 2]]);
cross[u_, v_] := Sum[GCD[u[[i]], v[[j]]], {i, 1, Length@u}, {j, 1, Length@v}];
a[n_] := If[n < 2, 0, s = 0; Do[s += permcount[p]*(2^(edges[p])*(2^cross[{1, 1}, p] + 2^cross[{2}, p])), {p, IntegerPartitions[n - 2]}]; s/(2(n - 2)!)];
Array[a, 20] (* Jean-François Alcover, Jul 07 2018, after Andrew Howroyd *)
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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) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i],v[j]))) + sum(i=1, #v, v[i]\2)}
cross(u,v) = {sum(i=1, #u, sum(j=1, #v, gcd(u[i],v[j])))}
a(n) = {if(n<2, 0, my(s=0); forpart(p=n-2, s+=permcount(p)*(2^(edges(p))*(2^cross([1,1],p) + 2^cross([2],p)))); s/(2*(n-2)!))} \\ Andrew Howroyd, May 03 2018
A339064
Number of unlabeled simple graphs with n edges rooted at two indistinguishable vertices.
Original entry on oeis.org
1, 3, 9, 28, 87, 276, 909, 3086, 10879, 39821, 151363, 597062, 2442044, 10342904, 45301072, 204895366, 955661003, 4590214994, 22675644514, 115068710553, 599149303234, 3197694533771, 17475917252052, 97712883807625, 558481251055893, 3260409769087068
Offset: 0
The a(1) = 3 cases correspond to a single edge which can be attached to zero, one or both of the roots.
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\\ See A339063 for G.
seq(n)={my(A=O(x*x^n)); Vec((G(2*n, x+A, [1, 1]) + G(2*n, x+A, [2]))/2)}
A361404
Triangle read by rows: T(n,k) is the number of graphs with loops on n unlabeled vertices with k loops.
Original entry on oeis.org
1, 1, 1, 2, 2, 2, 4, 6, 6, 4, 11, 20, 28, 20, 11, 34, 90, 148, 148, 90, 34, 156, 544, 1144, 1408, 1144, 544, 156, 1044, 5096, 13128, 20364, 20364, 13128, 5096, 1044, 12346, 79264, 250240, 472128, 580656, 472128, 250240, 79264, 12346
Offset: 0
Triangle begins:
1;
1, 1;
2, 2, 2;
4, 6, 6, 4;
11, 20, 28, 20, 11;
34, 90, 148, 148, 90, 34;
156, 544, 1144, 1408, 1144, 544, 156;
1044, 5096, 13128, 20364, 20364, 13128, 5096, 1044;
...
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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) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i], v[j]))) + sum(i=1, #v, v[i]\2)}
row(n) = {my(s=0); forpart(p=n, s+=permcount(p)*2^edges(p)*prod(i=1, #p, 1 + x^p[i])); Vecrev(s/n!)}
A340029
Number of unlabeled 2-connected graphs with n vertices rooted at a pair of indistinguishable vertices.
Original entry on oeis.org
0, 1, 1, 6, 37, 388, 6004, 148759, 5974184, 404509191, 47552739892, 9923861406343, 3735194287263442, 2565376853616300801, 3244070698095148283628, 7607050619214875184974489, 33269229977451262849539412860, 272689940536978851416633440863567
Offset: 1
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\\ See A004115 for graphsSeries and A339645 for combinatorial species functions.
blockGraphs(n)={my(gc=sLog(graphsSeries(n)), gcr=sPoint(gc)); intformal(x*sSolve( sLog( gcr/(x*sv(1)) ), gcr ), sv(1)) + sSolve(subst(gc, sv(1), 0), gcr)}
cycleIndexSeries(n)={sCartProd(blockGraphs(n), x^2 * symGroupCycleIndex(2) * symGroupSeries(n-2))}
{ my(N=15); Vec(OgfSeries(cycleIndexSeries(N)), -N) }
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