A322451 -id:A322451 - cp's OEIS frontend

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

N. J. A. Sloane

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

			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)

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).

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.

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.

(* 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 *)

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

Corrected and extended by Vladeta Jovovic

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

A289837 Number of cliques in the n-tetrahedral graph.

Original entry on oeis.org

1, 1, 2, 16, 76, 261, 757, 2003, 5035, 12286, 29426, 69554, 162670, 376923, 865971, 1973941, 4466853, 10040524, 22430584, 49829116, 110127536, 242254321, 530619937, 1157676711, 2516640751, 5452664426, 11777687182, 25367246038, 54492508610, 116769551831

Offset: 1

Eric W. Weisstein, Jul 13 2017

Here, "cliques" means complete subgraphs (not necessarily the largest).
Sequence extended to a(1) using formula. - Andrew Howroyd, Jul 18 2017
From Gus Wiseman, Jan 11 2019: (Start)
The n-tetrahedral graph has all 3-subsets of {1,...,n} as vertices, and two are connected iff they share two elements. So a(n) is the number of 3-uniform hypergraphs on n labeled vertices where every two edges have two vertices in common. For example, the a(4) = 16 hypergraphs are:
  {}
  {{1,2,3}}
  {{1,2,4}}
  {{1,3,4}}
  {{2,3,4}}
  {{1,2,3},{1,2,4}}
  {{1,2,3},{1,3,4}}
  {{1,2,3},{2,3,4}}
  {{1,2,4},{1,3,4}}
  {{1,2,4},{2,3,4}}
  {{1,3,4},{2,3,4}}
  {{1,2,3},{1,2,4},{1,3,4}}
  {{1,2,3},{1,2,4},{2,3,4}}
  {{1,2,3},{1,3,4},{2,3,4}}
  {{1,2,4},{1,3,4},{2,3,4}}
  {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}
The following are non-isomorphic representatives of the 7 unlabeled 3-uniform cliques on 6 vertices, and their multiplicities in the labeled case, which add up to a(6) = 261.
   1 X {}
  20 X {{1,2,3}}
  90 X {{1,3,4},{2,3,4}}
  60 X {{1,4,5},{2,4,5},{3,4,5}}
  60 X {{1,2,4},{1,3,4},{2,3,4}}
  15 X {{1,5,6},{2,5,6},{3,5,6},{4,5,6}}
  15 X {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..200
Eric Weisstein's World of Mathematics, Clique
Eric Weisstein's World of Mathematics, Tetrahedral Graph
Index entries for linear recurrences with constant coefficients, signature (11,-52,138,-225,231,-146,52,-8).

Cf. A055795 (maximal cliques), A287232 (independent vertex sets), A290056 (triangular graph).

Cf. A000665, A125791, A299471, A302374, A302394, A322451, A323293, A323296, A323298.

Table[(2^(n - 2) - n + 1) Binomial[n, 2] + Binomial[n, 3] +
  5 Binomial[n, 4] + 1, {n, 20}] (* Eric W. Weisstein, Jul 21 2017 *)
LinearRecurrence[{11, -52, 138, -225, 231, -146, 52, -8}, {1, 1, 2, 16, 76, 261, 757, 2003}, 20] (* Eric W. Weisstein, Jul 21 2017 *)
CoefficientList[Series[(1 - 10 x + 43 x^2 - 92 x^3 + 91 x^4 - 25 x^5 - 5 x^6 - 8 x^7)/((-1 + x)^5 (-1 + 2 x)^3), {x, 0, 20}], x] (* Eric W. Weisstein, Jul 21 2017 *)
stableSets[u_,Q_]:=If[Length[u]===0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r===w||Q[r,w]||Q[w,r]],Q]]]];
Table[Length[stableSets[Subsets[Range[n],{3}],Length[Intersection[#1,#2]]<=1&]],{n,6}] (* Gus Wiseman, Jan 11 2019 *)

a(n) = 1 + binomial(n,3) + (2^(n-2)-n+1)*binomial(n,2) + 5*binomial(n,4); \\ Andrew Howroyd, Jul 18 2017

Vec(x*(1 - 10*x + 43*x^2 - 92*x^3 + 91*x^4 - 25*x^5 - 5*x^6 - 8*x^7) / ((1 - x)^5*(1 - 2*x)^3) + O(x^40)) \\ Colin Barker, Jul 19 2017

a(n) = 1 + binomial(n,3) + (2^(n-2)-n+1)*binomial(n,2) + 5*binomial(n,4). - Andrew Howroyd, Jul 18 2017
a(n) = 11*a(n-1)-52*a(n-2)+138*a(n-3)-225*a(n-4)+231*a(n-5)-146*a(n-6)+52*a(n-7)-8*a(n-8). - Eric W. Weisstein, Jul 21 2017
From Colin Barker, Jul 19 2017: (Start)
G.f.: x*(1 - 10*x + 43*x^2 - 92*x^3 + 91*x^4 - 25*x^5 - 5*x^6 - 8*x^7) / ((1 - x)^5*(1 - 2*x)^3).
a(n) = (24 - (34+3*2^n)*n + (67+3*2^n)*n^2 - 38*n^3 + 5*n^4) / 24.
(End)
Binomial transform of A323294. - Gus Wiseman, Jan 11 2019

a(1)-a(5) and a(21)-a(30) from Andrew Howroyd, Jul 18 2017

A320395 Number of non-isomorphic 3-uniform multiset systems over {1,...,n}.

Original entry on oeis.org

1, 2, 10, 208, 45960, 287800704, 100103176111616, 3837878984050795692032, 32966965900633495618246298767360, 128880214965936601447070466061615999984402432, 464339910355487357558396669850788946402420533504952464572416

Offset: 0

Gus Wiseman, Dec 12 2018

nonn

			Non-isomorphic representatives of the a(2) = 10 multiset systems:
  {}
  {{111}}
  {{122}}
  {{111}{222}}
  {{112}{122}}
  {{112}{222}}
  {{122}{222}}
  {{111}{122}{222}}
  {{112}{122}{222}}
  {{111}{112}{122}{222}}

Andrew Howroyd, Table of n, a(n) for n = 0..25

The 2-uniform case is A000666. The case of sets (as opposed to multisets) is A000665. The case of labeled spanning sets is A302374, with unlabeled case A322451.

Cf. A000088, A000612, A003180, A070166, A301922, A317795, A319876.

Table[Sum[2^PermutationCycles[Ordering[Map[Sort,Select[Tuples[Range[n],3],OrderedQ]/.Rule@@@Table[{i,prm[[i]]},{i,n}],{1}]],Length],{prm,Permutations[Range[n]]}]/n!,{n,6}]

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}
rep(typ)={my(L=List(), k=0); for(i=1, #typ, k+=typ[i]; listput(L, k); while(#L0, u=vecsort(apply(f, u)); d=lex(u, v)); !d}
Q(perm)={my(t=0); forsubset([#perm+2, 3], v, t += can([v[1],v[2]-1,v[3]-2], t->perm[t])); t}
a(n)={my(s=0); forpart(p=n, s += permcount(p)*2^Q(rep(p))); s/n!} \\ Andrew Howroyd, Aug 26 2019

Terms a(9) and beyond from Andrew Howroyd, Aug 26 2019

A323293 Number of 3-uniform hypergraphs on n labeled vertices where no two edges have two vertices in common.

Original entry on oeis.org

1, 1, 1, 2, 5, 26, 271, 5596, 231577, 21286940, 4392750641, 2100400533176

Offset: 0

Gus Wiseman, Jan 10 2019

			The a(5) = 26 hypergraphs:
  {}
  {{1,2,3}}
  {{1,2,4}}
  {{1,2,5}}
  {{1,3,4}}
  {{1,3,5}}
  {{1,4,5}}
  {{2,3,4}}
  {{2,3,5}}
  {{2,4,5}}
  {{3,4,5}}
  {{1,2,3},{1,4,5}}
  {{1,2,3},{2,4,5}}
  {{1,2,3},{3,4,5}}
  {{1,2,4},{1,3,5}}
  {{1,2,4},{2,3,5}}
  {{1,2,4},{3,4,5}}
  {{1,2,5},{1,3,4}}
  {{1,2,5},{2,3,4}}
  {{1,2,5},{3,4,5}}
  {{1,3,4},{2,3,5}}
  {{1,3,4},{2,4,5}}
  {{1,3,5},{2,3,4}}
  {{1,3,5},{2,4,5}}
  {{1,4,5},{2,3,4}}
  {{1,4,5},{2,3,5}}
Non-isomorphic representatives of the 6 unlabeled 3-uniform hypertrees spanning 6 vertices where no two edges have two vertices in common, and their multiplicities in the labeled case which add up to a(6) = 271:
    1 X {}
   20 X {{1,2,3}}
   90 X {{1,2,5},{3,4,5}}
   10 X {{1,2,3},{4,5,6}}
  120 X {{1,3,5},{2,3,6},{4,5,6}}
   30 X {{1,2,4},{1,3,5},{2,3,6},{4,5,6}}

Essentially the same as A287232.

Cf. A000665, A025035, A125791, A190865, A289837, A302374, A302394, A319540, A320395, A322451, A323292-A323299.

stableSets[u_,Q_]:=If[Length[u]===0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r===w||Q[r,w]||Q[w,r]],Q]]]];
Table[Length[stableSets[Subsets[Range[n],{3}],Length[Intersection[#1,#2]]>1&]],{n,8}]

a(9) from Andrew Howroyd, Aug 14 2019

a(10) and a(11) (using A287232) from Joerg Arndt, Oct 12 2023

A323296 Number of 3-uniform hypergraphs spanning n labeled vertices where no two edges have exactly one vertex in common.

Original entry on oeis.org

1, 0, 0, 1, 11, 10, 25, 406, 4823, 15436, 72915, 895180, 11320441, 71777498, 519354927, 6155284240, 82292879425, 788821735656, 7772567489083, 98329764933354, 1400924444610675, 17424772471470490, 216091776292721021, 3035845122991962688, 46700545575567202903

Offset: 0

Gus Wiseman, Jan 11 2019

nonn

The only way to meet the requirements is to cover the vertices with zero or more disconnected 3-uniform hypergraphs with each edge having exactly two vertices in common (A323294). - Andrew Howroyd, Aug 18 2019

			The a(4) = 11:
  {{1,2,3},{1,2,4}}
  {{1,2,3},{1,3,4}}
  {{1,2,3},{2,3,4}}
  {{1,2,4},{1,3,4}}
  {{1,2,4},{2,3,4}}
  {{1,3,4},{2,3,4}}
  {{1,2,3},{1,2,4},{1,3,4}}
  {{1,2,3},{1,2,4},{2,3,4}}
  {{1,2,3},{1,3,4},{2,3,4}}
  {{1,2,4},{1,3,4},{2,3,4}}
  {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}
The following are non-isomorphic representatives of the 3 unlabeled 3-uniform hypergraphs spanning 7 vertices with no two edges having exactly one vertex in common, and their multiplicities in the labeled case, which add up to a(7) = 406.
  210 X {{1,2,3},{4,6,7},{5,6,7}}
  140 X {{1,2,3},{4,5,7},{4,6,7},{5,6,7}}
   21 X {{1,6,7},{2,6,7},{3,6,7},{4,6,7},{5,6,7}}
   35 X {{1,2,3},{4,5,6},{4,5,7},{4,6,7},{5,6,7}}

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A025035, A125791, A190865, A289837, A299471, A302374, A302394, A322451, A323293, A323294, A323297.

b:= n-> `if`(n<5, (n-2)*(2*n^2-6*n+3)/6, n/2)*(n-1):
a:= proc(n) option remember; `if`(n=0, 1, add(
      binomial(n-1, k-1)*b(k)*a(n-k), k=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 18 2019

stableSets[u_,Q_]:=If[Length[u]===0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r===w||Q[r,w]||Q[w,r]],Q]]]];
Table[Length[Select[stableSets[Subsets[Range[n],{3}],Length[Intersection[#1,#2]]==1&],Union@@#==Range[n]&]],{n,8}]

seq(n)={Vec(serlaplace(exp(-x^2/2 - x^3/3 + 5*x^4/24 + x^2*exp(x + O(x^(n-1)))/2)))} \\ Andrew Howroyd, Aug 18 2019

From Andrew Howroyd, Aug 18 2019: (Start)
Exponential transform of A323294.
E.g.f.: exp(-x^2/2 - x^3/3 + 5*x^4/24 + x^2*exp(x)/2). (End)

a(11) from Alois P. Heinz, Aug 12 2019

Terms a(12) and beyond from Andrew Howroyd, Aug 18 2019

A323299 Number of 3-uniform hypergraphs on n labeled vertices where every two edges have exactly one vertex in common.

Original entry on oeis.org

1, 1, 1, 2, 5, 26, 261, 3216, 19617, 80860, 262651, 737716, 1920821, 5013152, 14277485, 47610876, 186355041, 820625616, 3869589607, 19039193980, 96332399701, 499138921736, 2639262062801, 14234781051932, 78188865206145, 437305612997376, 2487692697142251

Offset: 0

Gus Wiseman, Jan 11 2019

nonn

			The a(5) = 26 hypergraphs:
  {}
  {{1,2,3}}
  {{1,2,4}}
  {{1,2,5}}
  {{1,3,4}}
  {{1,3,5}}
  {{1,4,5}}
  {{2,3,4}}
  {{2,3,5}}
  {{2,4,5}}
  {{3,4,5}}
  {{1,2,3},{1,4,5}}
  {{1,2,3},{2,4,5}}
  {{1,2,3},{3,4,5}}
  {{1,2,4},{1,3,5}}
  {{1,2,4},{2,3,5}}
  {{1,2,4},{3,4,5}}
  {{1,2,5},{1,3,4}}
  {{1,2,5},{2,3,4}}
  {{1,2,5},{3,4,5}}
  {{1,3,4},{2,3,5}}
  {{1,3,4},{2,4,5}}
  {{1,3,5},{2,3,4}}
  {{1,3,5},{2,4,5}}
  {{1,4,5},{2,3,4}}
  {{1,4,5},{2,3,5}}
The following are non-isomorphic representatives of the 10 unlabeled 3-uniform hypergraphs on 7 vertices where every two edges have exactly one vertex in common, and their multiplicities in the labeled case, which add up to a(7) = 3216.
    1 X {}
   35 X {{1,2,3}}
  315 X {{1,2,5},{3,4,5}}
  105 X {{1,2,7},{3,4,7},{5,6,7}}
  840 X {{1,3,5},{2,3,6},{4,5,6}}
  840 X {{1,4,5},{2,4,6},{3,4,7},{5,6,7}}
  210 X {{1,2,4},{1,3,5},{2,3,6},{4,5,6}}
  630 X {{1,4,5},{2,3,5},{2,4,6},{3,4,7},{5,6,7}}
  210 X {{1,3,6},{1,4,5},{2,3,5},{2,4,6},{3,4,7},{5,6,7}}
   30 X {{1,2,7},{1,3,6},{1,4,5},{2,3,5},{2,4,6},{3,4,7},{5,6,7}}

Cf. A025035, A125791, A190865, A289837, A299471, A302374, A302394, A322451, A323293, A323296, A323297, A323298.

stableSets[u_,Q_]:=If[Length[u]===0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r===w||Q[r,w]||Q[w,r]],Q]]]];
Table[Length[stableSets[Subsets[Range[n],{3}],Length[Intersection[#1,#2]]!=1&]],{n,8}]

Binomial transform of A323298.

Terms a(11) and beyond from Andrew Howroyd, Aug 14 2019

A319540 Number of unlabeled 3-uniform hypergraphs spanning n vertices such that every pair of vertices appears together in some block.

Original entry on oeis.org

1, 1, 0, 1, 2, 14, 964, 3908438

Offset: 0

Gus Wiseman, Jan 09 2019

			Non-isomorphic representatives of the a(5) = 14 hypergraphs:
              {{123}{145}{245}{345}}
            {{123}{124}{135}{245}{345}}
            {{123}{145}{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}}
        {{123}{124}{135}{145}{235}{245}{345}}
        {{124}{135}{145}{234}{235}{245}{345}}
        {{125}{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}}

The labeled case is A302394.

Cf. A000665, A006126, A006129, A038041, A299471, A301922, A302374, A306021, A320395, A322451.

a(6)-a(7) from Andrew Howroyd, Aug 17 2019

A323292 Number of 3-uniform hypergraphs spanning n labeled vertices where no two edges have two vertices in common.

Original entry on oeis.org

1, 0, 0, 1, 0, 15, 160, 4125, 193200, 19384225

Offset: 0

Gus Wiseman, Jan 10 2019

			The a(5) = 15 hypergraphs:
  {{1,2,3},{1,4,5}}
  {{1,2,3},{2,4,5}}
  {{1,2,3},{3,4,5}}
  {{1,2,4},{1,3,5}}
  {{1,2,4},{2,3,5}}
  {{1,2,4},{3,4,5}}
  {{1,2,5},{1,3,4}}
  {{1,2,5},{2,3,4}}
  {{1,2,5},{3,4,5}}
  {{1,3,4},{2,3,5}}
  {{1,3,4},{2,4,5}}
  {{1,3,5},{2,3,4}}
  {{1,3,5},{2,4,5}}
  {{1,4,5},{2,3,4}}
  {{1,4,5},{2,3,5}}
Non-isomorphic representatives of the 3 unlabeled 3-uniform hypergraphs spanning 6 vertices where no two edges have two vertices in common, and their multiplicities in the labeled case which add up to a(6) = 160:
   10 X {{1,2,3},{4,5,6}}
  120 X {{1,3,5},{2,3,6},{4,5,6}}
   30 X {{1,2,4},{1,3,5},{2,3,6},{4,5,6}}

Cf. A000665, A025035, A125791, A190865, A289837, A302374, A302394, A320395, A322451, A323293-A323299.

stableSets[u_,Q_]:=If[Length[u]===0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r===w||Q[r,w]||Q[w,r]],Q]]]];
Table[Length[Select[stableSets[Subsets[Range[n],{3}],Length[Intersection[#1,#2]]>=2&],Union@@#==Range[n]&]],{n,6}]

Inverse binomial transform of A323293. - Andrew Howroyd, Aug 14 2019

a(9) from Andrew Howroyd, Aug 14 2019

A323294 Number of 3-uniform hypergraphs spanning n labeled vertices where every two edges have two vertices in common.

Original entry on oeis.org

1, 0, 0, 1, 11, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946, 990, 1035, 1081, 1128, 1176, 1225, 1275, 1326, 1378, 1431

Offset: 0

Gus Wiseman, Jan 10 2019

nonn

			The a(4) = 11 hypergraphs:
  {{1,2,3},{1,2,4}}
  {{1,2,3},{1,3,4}}
  {{1,2,3},{2,3,4}}
  {{1,2,4},{1,3,4}}
  {{1,2,4},{2,3,4}}
  {{1,3,4},{2,3,4}}
  {{1,2,3},{1,2,4},{1,3,4}}
  {{1,2,3},{1,2,4},{2,3,4}}
  {{1,2,3},{1,3,4},{2,3,4}}
  {{1,2,4},{1,3,4},{2,3,4}}
  {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Cf. A000217, A000665, A025035, A125791, A161680, A289837, A302374, A302394, A319540, A320395, A322451, A323292-A323299.

stableSets[u_,Q_]:=If[Length[u]===0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r===w||Q[r,w]||Q[w,r]],Q]]]];
Table[Length[Select[stableSets[Subsets[Range[n],{3}],Length[Intersection[#1,#2]]<=1&],Union@@#==Range[n]&]],{n,10}]

seq(n)={Vec(serlaplace(1 - x^2/2 - x^3/3 + 5*x^4/24 + x^2*exp(x + O(x^(n-1)))/2))} \\ Andrew Howroyd, Aug 18 2019

a(n) = binomial(n,2) for n >= 5. - Gus Wiseman, Jan 16 2019
Binomial transform is A289837. - Gus Wiseman, Jan 16 2019
a(n) = A000217(n-1) for n >= 5. - Alois P. Heinz, Jan 24 2019
E.g.f.: 1 - x^2/2 - x^3/3 + 5*x^4/24 + x^2*exp(x)/2. - Andrew Howroyd, Aug 18 2019

A323298 Number of 3-uniform hypergraphs spanning n labeled vertices where every two edges have exactly one vertex in common.

Original entry on oeis.org

1, 0, 0, 1, 0, 15, 150, 1815, 0, 945, 0, 10395, 0, 135135, 0, 2027025, 0, 34459425, 0, 654729075, 0, 13749310575, 0, 316234143225, 0, 7905853580625, 0, 213458046676875, 0, 6190283353629375, 0, 191898783962510625, 0, 6332659870762850625, 0, 221643095476699771875

Offset: 0

Gus Wiseman, Jan 11 2019

nonn

The only way to cover more than 7 vertices is with edges all having a single common vertex. For the special cases of n = 6 or n = 7, there are also covers without a common vertex. - Andrew Howroyd, Aug 15 2019

			The a(5) = 15 hypergraphs:
  {{1,4,5},{2,3,5}}
  {{1,4,5},{2,3,4}}
  {{1,3,5},{2,4,5}}
  {{1,3,5},{2,3,4}}
  {{1,3,4},{2,4,5}}
  {{1,3,4},{2,3,5}}
  {{1,2,5},{3,4,5}}
  {{1,2,5},{2,3,4}}
  {{1,2,5},{1,3,4}}
  {{1,2,4},{3,4,5}}
  {{1,2,4},{2,3,5}}
  {{1,2,4},{1,3,5}}
  {{1,2,3},{3,4,5}}
  {{1,2,3},{2,4,5}}
  {{1,2,3},{1,4,5}}
The following are non-isomorphic representatives of the 5 unlabeled 3-uniform hypergraphs spanning 7 vertices in which every two edges have exactly one vertex in common, and their multiplicities in the labeled case, which add up to a(7) = 1815.
  105 X {{1,2,7},{3,4,7},{5,6,7}}
  840 X {{1,4,5},{2,4,6},{3,4,7},{5,6,7}}
  630 X {{1,4,5},{2,3,5},{2,4,6},{3,4,7},{5,6,7}}
  210 X {{1,3,6},{1,4,5},{2,3,5},{2,4,6},{3,4,7},{5,6,7}}
   30 X {{1,2,7},{1,3,6},{1,4,5},{2,3,5},{2,4,6},{3,4,7},{5,6,7}}
From _Andrew Howroyd_, Aug 15 2019: (Start)
The following are non-isomorphic representatives of the 2 unlabeled 3-uniform hypergraphs spanning 6 vertices in which every two edges have exactly one vertex in common, and their multiplicities in the labeled case, which add up to a(6) = 150.
    120 X {{1,2,3},{1,4,5},{3,5,6}}
     30 X {{1,2,3},{1,4,5},{3,5,6},{2,4,6}}
(End)

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A001147, A025035, A125791, A190865, A289837, A299471, A302374, A302394, A322451, A323293, A323296, A323297, A323299.

stableSets[u_,Q_]:=If[Length[u]===0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r===w||Q[r,w]||Q[w,r]],Q]]]];
Table[Length[Select[stableSets[Subsets[Range[n],{3}],Length[Intersection[#1,#2]]!=1&],Union@@#==Range[n]&]],{n,10}]

a(n)={if(n%2, if(n<=3, n==3, if(n==7, 1815, n!/(2^(n\2)*(n\2)!))), if(n==6, 150, n==0))} \\ Andrew Howroyd, Aug 15 2019

a(2*n) = 0 for n > 3; a(2*n-1) = A001147(n) for n > 4. - Andrew Howroyd, Aug 15 2019

Terms a(13) and beyond from Andrew Howroyd, Aug 15 2019

cp's OEIS Frontend

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

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A289837 Number of cliques in the n-tetrahedral graph.

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A320395 Number of non-isomorphic 3-uniform multiset systems over {1,...,n}.

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A323293 Number of 3-uniform hypergraphs on n labeled vertices where no two edges have two vertices in common.

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A323296 Number of 3-uniform hypergraphs spanning n labeled vertices where no two edges have exactly one vertex in common.

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A323299 Number of 3-uniform hypergraphs on n labeled vertices where every two edges have exactly one vertex in common.

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A319540 Number of unlabeled 3-uniform hypergraphs spanning n vertices such that every pair of vertices appears together in some block.

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A323292 Number of 3-uniform hypergraphs spanning n labeled vertices where no two edges have two vertices in common.

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