cp's OEIS Frontend

A290428 Array read by antidiagonals: T(n,k) is the number of graphs with n edges and k vertices, allowing loops and multi-edges.

%I A290428 #31 Jan 08 2021 08:47:51
%S A290428 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,
%T A290428 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,
%U A290428 198,287,152,25,1,0,1,2,7,23,78,245,550,606,242,30,1,0
%N A290428 Array read by antidiagonals: T(n,k) is the number of graphs with n edges and k vertices, allowing loops and multi-edges.
%C A290428 Variant of A138107, here for non-directed edges.
%H A290428 Andrew Howroyd, <a href="/A290428/b290428.txt">Table of n, a(n) for n = 0..1325</a>
%H A290428 R. J. Mathar, <a href="http://arxiv.org/abs/1709.09000">Statistics on Small Graphs</a>, arXiv:1709.09000 [math.CO], 2017. See Table 59.
%e A290428 1   1   1   1    1    1    1   1   1...
%e A290428 0   1   2   2    2    2    2   2   2...
%e A290428 0   1   4   6    7    7    7   7   7...
%e A290428 0   1   6  14   20   22   23  23  23...
%e A290428 0   1   9  28   53   69   76  78  79...
%e A290428 0   1  12  52  125  198  245 264 271...
%e A290428 0   1  16  93  287  550  782 915 973...
%e A290428 0   1  20 152  606 1441 2392
%e A290428 0   1  25 242 1226 3611
%t A290428 rows = 12;
%t A290428 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];
%t A290428 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]}];
%t A290428 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 A290428 T[0, _] = 1; T[_, 0] = 0;
%t A290428 T[n_, k_] := col[k][[n + 1]];
%t A290428 Table[T[n-k, k], {n, 0, rows-1}, {k, n, 0, -1}] // Flatten (* _Jean-François Alcover_, Jan 08 2021, after _Andrew Howroyd_ *)
%o A290428 (PARI)
%o A290428 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 A290428 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)))}
%o A290428 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)!)))}
%o A290428 { my(A=T(8)); for(n=1, #A, print(A[n,])) } \\ _Andrew Howroyd_, Oct 22 2019
%Y A290428 Cf. A050531 (column 3), A050532 (column 4), A138107, A098568 (vertex-labeled).
%K A290428 nonn,tabl
%O A290428 0,8
%A A290428 _R. J. Mathar_, Jul 31 2017