A304074
Number of simple connected graphs with n nodes rooted at a pair of distinguished vertices.
Original entry on oeis.org
0, 1, 4, 23, 162, 1549, 21090, 446061, 15673518, 961338288, 105752617892, 21155707801451, 7757777336382702, 5245054939576054088, 6571185585793205495484, 15325133281701584879975433, 66813349775478836190531605234, 546646811841381587823502759339055
Offset: 1
a(3)=4: one choice to mark two roots in the triangular graph; one choice to mark the two leaves in the linear graph; two choices to mark the center node and a leave (1st root in the center or 2nd root in the center) in the linear graph.
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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])))}
S(n, r)={my(t=#r+1); vector(n+1, n, if(nAndrew Howroyd, Sep 07 2019
A304069
Number of simple graphs on n vertices rooted at one oriented edge.
Original entry on oeis.org
0, 1, 4, 20, 120, 996, 12208, 241520, 8171936, 491317640, 53489987584, 10642774095040, 3891541970165760, 2627082058057474240, 3288629181834544457216, 7666328470407977450185984, 33415367571344085375628748800, 273361007807597539567353971109952
Offset: 1
a(3)=4: no contribution from the graph with 3 isolated nodes. 1 case of the connected graph with 2 nodes and an isolated node. 2 cases of the linear graph with 3 nodes (orientation either towards or away from the middle node). 1 case of the triangular graph.
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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[#, 2]& /@ v];
a[n_] := If[n < 2, 0, s = 0; Do[s += permcount[p]*(2^(2*Length[p] + edges[p])), {p, IntegerPartitions[n - 2]}]; s/(n - 2)!];
Array[a, 18] (* Jean-François Alcover, Jul 03 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)}
a(n)= {if(n<2, 0, my(s=0); forpart(p=n-2, s+=permcount(p)*(2^(2*#p+edges(p)))); s/(n-2)!)} \\ Andrew Howroyd, May 06 2018
A339044
Number of unlabeled connected simple graphs with n edges rooted at one oriented edge.
Original entry on oeis.org
1, 2, 6, 18, 57, 188, 651, 2336, 8719, 33741, 135185, 559908, 2394326, 10557283, 47943126, 223987316, 1075455181, 5301593544, 26807904317, 138924912857, 737220195148, 4002876571636, 22221898966507, 126042573704637, 729944250603862, 4313430995825272
Offset: 1
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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, [])/(1+x))}
A304073
Number of simple connected graphs with n nodes rooted at one oriented non-edge.
Original entry on oeis.org
0, 0, 1, 8, 67, 701, 10047, 218083, 7758105, 478466565, 52762737260, 10566937121191, 3876933205880431, 2621875289142578194, 3285187439267316978728, 7662096100649423384254265, 33405651855362295512020765765, 273319227135047244053866187609854
Offset: 1
a(3)=1: no contribution from the triangle graph; one case of joining the leaves of the linear graph.
a(4)=8: we start from the 6 cases of non-oriented non-edges of A304071 and note two geometries where the orientation makes a difference: for the triangular graph with a protruding edge the orientation matters (to or from the leaf), and also for the linear graph with 4 nodes (to or from the leaf).
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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])))}
S(n, r)={my(t=#r+1); vector(n+1, n, if(nAndrew Howroyd, Sep 07 2019
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