A290428 Array read by antidiagonals: T(n,k) is the number of graphs with n edges and k vertices, allowing loops and multi-edges.
1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 2, 4, 1, 0, 1, 2, 6, 6, 1, 0, 1, 2, 7, 14, 9, 1, 0, 1, 2, 7, 20, 28, 12, 1, 0, 1, 2, 7, 22, 53, 52, 16, 1, 0, 1, 2, 7, 23, 69, 125, 93, 20, 1, 0, 1, 2, 7, 23, 76, 198, 287, 152, 25, 1, 0, 1, 2, 7, 23, 78, 245, 550, 606, 242, 30, 1, 0
Offset: 0
Examples
1 1 1 1 1 1 1 1 1... 0 1 2 2 2 2 2 2 2... 0 1 4 6 7 7 7 7 7... 0 1 6 14 20 22 23 23 23... 0 1 9 28 53 69 76 78 79... 0 1 12 52 125 198 245 264 271... 0 1 16 93 287 550 782 915 973... 0 1 20 152 606 1441 2392 0 1 25 242 1226 3611
Links
- Andrew Howroyd, Table of n, a(n) for n = 0..1325
- R. J. Mathar, Statistics on Small Graphs, arXiv:1709.09000 [math.CO], 2017. See Table 59.
Programs
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Mathematica
rows = 12; 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_, t_] := Product[g = GCD[v[[i]], v[[j]] ]; t[v[[i]]*v[[j]]/g]^g, {i, 2, Length[v]}, {j, 1, i - 1}]*Product[c = v[[i]]; t[c]^Quotient[c + 1, 2]*If[OddQ[c], 1, t[c/2]], {i, 1, Length[v]}]; col[k_] := col[k] = Module[{s = O[x]^rows}, Do[s += permcount[p]*1/edges[p, 1 - x^# + O[x]^rows&], {p, IntegerPartitions[k]}]; s/k!] // CoefficientList[#, x]&; T[0, ] = 1; T[, 0] = 0; T[n_, k_] := col[k][[n + 1]]; Table[T[n-k, k], {n, 0, rows-1}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Jan 08 2021, after Andrew Howroyd *) -
PARI
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)^g )) * prod(i=1, #v, my(c=v[i]); t(c)^((c+1)\2)*if(c%2, 1, t(c/2)))} T(m, n=m) = {Mat(vector(n+1, n, my(s=O(x*x^m)); forpart(p=n-1, s+=permcount(p)*1/edges(p,i->1-x^i+O(x*x^m))); Col(s/(n-1)!)))} { my(A=T(8)); for(n=1, #A, print(A[n,])) } \\ Andrew Howroyd, Oct 22 2019
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