A000665 Number of 3-uniform hypergraphs on n unlabeled nodes, or equivalently number of relations with 3 arguments on n nodes.

1, 1, 1, 2, 5, 34, 2136, 7013320, 1788782616656, 53304527811667897248, 366299663432194332594005123072, 1171638318502989084030402509596875836036608, 3517726593606526072882013063011594224625680712384971214848

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice

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

Examples

			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)
		

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.

Crossrefs

Row sums of A092337. Spanning 3-uniform hypergraphs are counted by A322451.

Cf. A000088, A006129, A038041, A299471, A301922, A302374, A302394, A306017, A306021, A320395.

Column k=3 of A309858.

Programs

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 *)

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

Extensions

Corrected and extended by Vladeta Jovovic

a(0)=1 prepended and a(12) from Andrew Howroyd, Dec 11 2018