OFFSET
0,4
COMMENTS
The Qian reference has one incorrect term. The formula given in corollary 2.6 also contains a minor error. The second summation needs to be over p_i*p_j*p_h/lcm(p_i, p_j, p_h) rather than gcd(p_i, p_j, p_h)^2. - Andrew Howroyd, Dec 11 2018
REFERENCES
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 231.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..28 (first 26 terms from Andrew Howroyd)
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Victor Falgas-Ravry and Emil R. Vaughan, On applications of Razborov's flag algebra calculus to extremal 3-graph theory, arXiv preprint arXiv:1110.1623 [math.CO], 2011.
W. Oberschelp, Kombinatorische Anzahlbestimmungen in Relationen, Math. Ann., 174 (1967), 53-78.
E. M. Palmer, On the number of n-plexes, Discrete Math., 6 (1973), 377-390.
Jianguo Qian, Enumeration of unlabeled uniform hypergraphs, Discrete Math. 326 (2014), 66--74. MR3188989.
EXAMPLE
From Gus Wiseman, Dec 13 2018: (Start)
Non-isomorphic representatives of the a(5) = 34 hypergraphs:
{}
{{123}}
{{125}{345}}
{{134}{234}}
{{123}{245}{345}}
{{124}{134}{234}}
{{135}{245}{345}}
{{145}{245}{345}}
{{123}{124}{134}{234}}
{{123}{145}{245}{345}}
{{124}{135}{245}{345}}
{{125}{135}{245}{345}}
{{134}{235}{245}{345}}
{{145}{235}{245}{345}}
{{123}{124}{135}{245}{345}}
{{123}{145}{235}{245}{345}}
{{124}{134}{235}{245}{345}}
{{134}{145}{235}{245}{345}}
{{135}{145}{235}{245}{345}}
{{145}{234}{235}{245}{345}}
{{123}{124}{134}{235}{245}{345}}
{{123}{134}{145}{235}{245}{345}}
{{123}{145}{234}{235}{245}{345}}
{{124}{135}{145}{235}{245}{345}}
{{125}{135}{145}{235}{245}{345}}
{{135}{145}{234}{235}{245}{345}}
{{123}{124}{135}{145}{235}{245}{345}}
{{124}{135}{145}{234}{235}{245}{345}}
{{125}{135}{145}{234}{235}{245}{345}}
{{134}{135}{145}{234}{235}{245}{345}}
{{123}{124}{135}{145}{234}{235}{245}{345}}
{{125}{134}{135}{145}{234}{235}{245}{345}}
{{124}{125}{134}{135}{145}{234}{235}{245}{345}}
{{123}{124}{125}{134}{135}{145}{234}{235}{245}{345}}
(End)
MATHEMATICA
(* about 85 seconds on a laptop computer *)
Needs["Combinatorica`"]; Table[A = Subsets[Range[n], {3}]; CycleIndex[Replace[Map[Sort, System`PermutationReplace[A, SymmetricGroup[n]], {2}], Table[A[[i]] -> i, {i, 1, Length[A]}], 2], s] /. Table[s[i] -> 2, {i, 1, Binomial[n, 3]}], {n, 1, 8}] (* Geoffrey Critzer, Oct 28 2015 *)
Table[Sum[2^PermutationCycles[Ordering[Map[Sort, Subsets[Range[n], {3}]/.Rule@@@Table[{i, prm[[i]]}, {i, n}], {1}]], Length], {prm, Permutations[Range[n]]}]/n!, {n, 8}] (* Gus Wiseman, Dec 13 2018 *)
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[p_] := Sum[Ceiling[(p[[i]] - 1)*((p[[i]] - 2)/6)], {i, 1, Length[p]}] + Sum[Sum[c = p[[i]]; d = p[[j]]; GCD[c, d]*(c + d - 2 + Mod[(c - d)/GCD[c, d], 2])/2 + Sum[c*d*p[[k]]/LCM[c, d, p[[k]]], {k, 1, j - 1}], {j, 1, i - 1}], {i, 2, Length[p]}];
a[n_] := Module[{s = 0}, Do[s += permcount[p]*2^edges[p], {p, IntegerPartitions[n]}]; s/n!];
a /@ Range[0, 12] (* 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(p)={sum(i=1, #p, ceil((p[i]-1)*(p[i]-2)/6)) + sum(i=2, #p, sum(j=1, i-1, my(c=p[i], d=p[j]); gcd(c, d)*(c + d - 2 + (c-d)/gcd(c, d)%2)/2 + sum(k=1, j-1, c*d*p[k]/lcm(lcm(c, d), p[k]))))}
a(n) = {my(s=0); forpart(p=n, s+=permcount(p)*2^edges(p)); s/n!} \\ Andrew Howroyd, Dec 11 2018
CROSSREFS
Column k=3 of A309858.
KEYWORD
nonn,nice
AUTHOR
EXTENSIONS
Corrected and extended by Vladeta Jovovic
a(0)=1 prepended and a(12) from Andrew Howroyd, Dec 11 2018
STATUS
approved