OFFSET
0,5
COMMENTS
T(n,k) is also the number of graphs with n nodes and k edges which may contain loops. (Delete the root node and change every edge leading to it into a loop.)
T(n,k) is also the number of symmetric relations with n points and k relations.
REFERENCES
E. Palmer and F. Harary, Graphical Enumeration, Academic Press, 1973.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1560
Marko R. Riedel, Number of distinct connected digraphs
Eric Weisstein's World of Mathematics, Rooted Graphs
EXAMPLE
Triangle begins:
1;
1, 1;
1, 2, 2, 1;
1, 2, 4, 6, 4, 2, 1;
1, 2, 5, 11, 17, 18, 17, 11, 5, 2, 1; <- gives either the numbers of rooted graphs on 5 nodes, or the numbers of graphs with loops on 4 nodes; with from 0 to 10 edges
1, 2, 5, 13, 29, 52, 76, 94, 94, 76, 52, 29, 13, 5, 2, 1;
...
MATHEMATICA
Join[{{1}, {1, 1}}, CoefficientList[Table[CycleIndex[Join[PairGroup[SymmetricGroup[n]], Permutations[Range[Binomial[n, 2]+1, Binomial[n, 2]+n]], 2], s]/.Table[s[i]->1+x^i, {i, 1, n^2-n}], {n, 2, 7}], x]]//Grid (* Geoffrey Critzer, Oct 01 2012 *)
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[Product[g = GCD[v[[i]], v[[j]]]; t[v[[i]]*v[[j]]/g]^g, {j, 1, i - 1}], {i, 2, Length[v]}]*Product[c = v[[i]]; t[c]^Quotient[c + 1, 2]*If[OddQ[c], 1, t[c/2]], {i, 1, Length[v]}];
row[n_] := Module[{s = 0}, Do[s += permcount[p]*edges[p, 1 + x^# &], {p, IntegerPartitions[n]}]; s/n!] // Expand // CoefficientList[#, x] &
row /@ Range[0, 7] // Flatten (* Jean-François Alcover, Jan 07 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)))}
G(n, A=0) = {my(s=0); forpart(p=n, s+=permcount(p)*edges(p, i->1+x^i+A)); s/n!}
{ for(n=0, 7, print(Vecrev(G(n)))) } \\ Andrew Howroyd, Oct 23 2019, updated Jan 09 2024
CROSSREFS
KEYWORD
nonn,tabf,nice
AUTHOR
Vladeta Jovovic and Eric W. Weisstein, Apr 23 2002
EXTENSIONS
Offset changed by Andrew Howroyd, Oct 23 2019
STATUS
approved