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

0,8

Comments

Variant of A138107, here for non-directed edges.

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.

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A050531 (column 3), A050532 (column 4), A138107, A098568 (vertex-labeled).

Sequence in context: A022958 A023444 A248157 * A136868 A276066 A145895

Adjacent sequences: A290425 A290426 A290427 * A290429 A290430 A290431

Keyword

nonn,tabl

Author

R. J. Mathar, Jul 31 2017

Status

approved