A301922 -id:A301922 - 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

A299471 Regular triangle where T(n,k) is the number of labeled k-uniform hypergraphs spanning n vertices.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 41, 11, 1, 1, 768, 958, 26, 1, 1, 27449, 1042642, 32596, 57, 1, 1, 1887284, 34352419335, 34359509614, 2096731, 120, 1, 1, 252522481, 72057319189324805, 1180591620442534312297, 72057594021152435, 268434467, 247, 1, 1, 66376424160

Offset: 1

Gus Wiseman, Jun 18 2018

			Triangle begins:
  1;
  1,     1;
  1,     4,       1;
  1,    41,      11,     1;
  1,   768,     958,    26,  1;
  1, 27449, 1042642, 32596, 57, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..91 (rows 1..13)
Wikipedia, Hypergraph

Columns 1..4 are A000012, A006129, A302374, A302396.
Row sums are A306021.
The unlabeled version is A301922.
The connected version is A299354.
Cf. A000005, A001315, A006126, A038041, A298422, A298426, A306017, A306018, A306019, A306020.

Table[Sum[(-1)^(n-d)*Binomial[n,d]*2^Binomial[d,k],{d,0,n}],{n,10},{k,n}]

T(n, k) = sum(d = 0, n, (-1)^(n-d)*binomial(n,d)*2^binomial(d,k)) \\ Andrew Howroyd, Jan 16 2024

T(n, k) = Sum_{d = 0..n} (-1)^(n-d)*binomial(n,d)*2^binomial(d,k).

A309858 Number A(n,k) of k-uniform hypergraphs on n unlabeled nodes; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

2, 1, 2, 1, 2, 2, 1, 1, 3, 2, 1, 1, 2, 4, 2, 1, 1, 1, 4, 5, 2, 1, 1, 1, 2, 11, 6, 2, 1, 1, 1, 1, 5, 34, 7, 2, 1, 1, 1, 1, 2, 34, 156, 8, 2, 1, 1, 1, 1, 1, 6, 2136, 1044, 9, 2, 1, 1, 1, 1, 1, 2, 156, 7013320, 12346, 10, 2, 1, 1, 1, 1, 1, 1, 7, 7013320, 1788782616656, 274668, 11, 2

Offset: 0

Alois P. Heinz, Aug 20 2019

See A000088 and A000665 for more references.

			Square array A(n,k) begins:
  2, 1,    1,       1,       1,    1, 1, 1, ...
  2, 2,    1,       1,       1,    1, 1, 1, ...
  2, 3,    2,       1,       1,    1, 1, 1, ...
  2, 4,    4,       2,       1,    1, 1, 1, ...
  2, 5,   11,       5,       2,    1, 1, 1, ...
  2, 6,   34,      34,       6,    2, 1, 1, ...
  2, 7,  156,    2136,     156,    7, 2, 1, ...
  2, 8, 1044, 7013320, 7013320, 1044, 8, 2, ...

Alois P. Heinz, Antidiagonals n = 0..20, flattened
Jianguo Qian, Enumeration of unlabeled uniform hypergraphs, Discrete Math. 326 (2014), 66--74. MR3188989.
Wikipedia, Hypergraph

Columns k=0..10 give: A007395, A000027, A000088, A000665, A051240, A051249, A309860, A309861, A309862, A309863, A309864.

Cf. A301922, A309865 (the same as triangle).

g:= (l, i, n)-> `if`(i=0, `if`(n=0, [[]], []), [seq(map(x->
     [x[], j], g(l, i-1, n-j))[], j=0..min(l[i], n))]):
h:= (p, v)-> (q-> add((s-> add(`if`(andmap(i-> irem(k[i], p[i]
     /igcd(t, p[i]))=0, [$1..q]), mul((m-> binomial(m, k[i]*m
     /p[i]))(igcd(t, p[i])), i=1..q), 0), t=1..s)/s)(ilcm(seq(
    `if`(k[i]=0, 1, p[i]), i=1..q))), k=g(p, q, v)))(nops(p)):
b:= (n, i, l, v)-> `if`(n=0 or i=1, 2^((p-> h(p, v))([l[], 1$n]))
     /n!, add(b(n-i*j, i-1, [l[], i$j], v)/j!/i^j, j=0..n/i)):
A:= proc(n, k) option remember; `if`(k>n, 1,
     `if`(k>n-k, A(n, n-k), b(n$2, [], k)))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);

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(n, k, perm)={my(t=0); forsubset([n, k], v, t += can(Vec(v), t->perm[t])); t}
T(n, k)={my(s=0); forpart(p=n, s += permcount(p)*2^Q(n, k, rep(p))); s/n!} \\ Andrew Howroyd, Aug 22 2019

A(n,k) = A(n,n-k) for 0 <= k <= n.

A(n,k) - A(n-1,k) = A301922(n,k) for n,k >= 1.

A322451 Number of unlabeled 3-uniform hypergraphs spanning n vertices.

Original entry on oeis.org

1, 0, 0, 1, 3, 29, 2102, 7011184, 1788775603336, 53304526022885280592, 366299663378889804782337225824, 1171638318502622784366970315264281830913536, 3517726593606524901243694560022510194223171115509135178240

Offset: 0

Gus Wiseman, Dec 09 2018

nonn

3-uniform means that every edge consists of 3 vertices. - Brendan McKay, Sep 03 2023

			Non-isomorphic representatives of the a(5) = 29 hypergraphs:
  {{125}{345}}
  {{123}{245}{345}}
  {{135}{245}{345}}
  {{145}{245}{345}}
  {{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}}

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

First differences of A000665.

Cf. A006126, A006129, A038041, A299471, A301922, A302374, A302394, A306017, A306021.

a(12) from Andrew Howroyd, Dec 15 2018

Name corrected by Brendan McKay, Sep 03 2023

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

A051240 Number of 4-uniform hypergraphs on n unlabeled nodes, or equivalently pure 3-dimensional simplicial complexes on n unlabeled nodes.

Original entry on oeis.org

1, 1, 1, 1, 2, 6, 156, 7013320, 29281354514767168, 234431745534048922731115555415680, 453456943706240925312368435237969462391555337386758635520

Offset: 0

Vladeta Jovovic

V. Jovovic, On the number of m-place relations (in Russian), Logiko-algebraicheskie konstruktsii, Tver, 1992, 59-66.
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..18 (terms n=1..16 from Andrew Howroyd)
E. M. Palmer, On the number of n-plexes, Discrete Math., 6 (1973), 377-390. [The numbers are incorrect]

Cf. A000665, A092337, A301922.

Column k=4 of A309858.

a(n) = 1 + Sum_{i=1..n} A301922(i, 4). - Andrew Howroyd, Aug 09 2019

a(0)=1 prepended by Alois P. Heinz, Aug 20 2019

A301481 Number of unlabeled uniform hypergraphs spanning n vertices.

Original entry on oeis.org

1, 1, 2, 4, 12, 58, 2381, 14026281, 29284932065996445, 468863491068204425232922367150021, 1994324729204021501147398087008429476673379600542622915802043462326345

Offset: 0

Gus Wiseman, Jun 19 2018

nonn

A hypergraph is uniform if all edges have the same size.

			Non-isomorphic representatives of the a(4) = 12 hypergraphs:
  {{1,2,3,4}}
  {{1,2},{3,4}}
  {{1},{2},{3},{4}}
  {{1,3,4},{2,3,4}}
  {{1,3},{2,4},{3,4}}
  {{1,4},{2,4},{3,4}}
  {{1,2,4},{1,3,4},{2,3,4}}
  {{1,2},{1,3},{2,4},{3,4}}
  {{1,4},{2,3},{2,4},{3,4}}
  {{1,3},{1,4},{2,3},{2,4},{3,4}}
  {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}
  {{1,2},{1,3},{1,4},{2,3},{2,4},{3,4}}

Alois P. Heinz, Table of n, a(n) for n = 0..14 (first 13 terms from Andrew Howroyd)

Row sums of A301922.

Cf. A003465, A038041, A055621, A298422, A301920, A306017-A306021.

\\ see A301922 for U(n,k).
a(n)={ if(n==0, 1, sum(k=1, n, U(n,k)-U(n-1,k))) } \\ Andrew Howroyd, Aug 10 2019

Terms a(6) and beyond from Andrew Howroyd, Aug 09 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

A301924 Regular triangle where T(n,k) is the number of unlabeled k-uniform connected hypergraphs spanning n vertices.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 6, 3, 1, 0, 21, 29, 4, 1, 0, 112, 2101, 150, 5, 1, 0, 853, 7011181, 7013164, 1037, 6, 1, 0, 11117, 1788775603301, 29281354507753847, 1788782615612, 12338, 7, 1, 0, 261080, 53304526022885278403, 234431745534048893449761040648508, 234431745534048922729326772799024, 53304527811667884902, 274659, 8, 1

Offset: 1

Gus Wiseman, Jun 19 2018

			Triangle begins:
   1
   0    1
   0    2       1
   0    6       3       1
   0   21      29       4    1
   0  112    2101     150    5 1
   0  853 7011181 7013164 1037 6 1
   ...
The T(4,2) = 6 hypergraphs:
  {{1,3},{2,4},{3,4}}
  {{1,4},{2,4},{3,4}}
  {{1,2},{1,3},{2,4},{3,4}}
  {{1,4},{2,3},{2,4},{3,4}}
  {{1,3},{1,4},{2,3},{2,4},{3,4}}
  {{1,2},{1,3},{1,4},{2,3},{2,4},{3,4}}

Row sums are A301920.
Columns k=2..3 are A001349(n > 1), A003190(n > 1).
Cf. A003465, A006129, A038041, A055621, A298422, A299353, A299354, A299471, A301481, A301922, A306017-A306021, A309858.

InvEulerT(v)={my(p=log(1+x*Ser(v))); dirdiv(vector(#v,n,polcoeff(p,n)), vector(#v,n,1/n))}
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(n, k, perm)={my(t=0); forsubset([n, k], v, t += can(Vec(v), t->perm[t])); t}
U(n, k)={my(s=0); forpart(p=n, s += permcount(p)*2^Q(n, k, rep(p))); s/n!}
A(n)={Mat(vector(n, k, InvEulerT(vector(n,i,U(i,k)-U(i-1,k)))~))}
{ my(T=A(8)); for(n=1, #T, print(T[n,1..n])) } \\ Andrew Howroyd, Aug 26 2019

Column k is the inverse Euler transform of column k of A301922. - Andrew Howroyd, Aug 26 2019

Terms a(16) and beyond from Andrew Howroyd, Aug 26 2019

A003190 Number of connected 2-plexes.

Original entry on oeis.org

1, 0, 1, 3, 29, 2101, 7011181, 1788775603301, 53304526022885278403, 366299663378889804782330207902, 1171638318502622784366970315262493034215728, 3517726593606524901243694560022510194169866584119717555335

Offset: 1

N. J. A. Sloane

The Palmer reference (incorrectly) has a(7)=7011349, a(8)=1788775603133, a(9)=53304526022885278659. - Sean A. Irvine, Mar 05 2015

Also connected 3-uniform hypergraphs on n vertices. - Gus Wiseman, Feb 23 2019

			From _Gus Wiseman_, Feb 23 2019: (Start)
Non-isomorphic representatives of the a(5) = 29 2-plexes:
  {{125}{345}}
  {{123}{245}{345}}
  {{135}{245}{345}}
  {{145}{245}{345}}
  {{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)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
E. M. Palmer, On the number of n-plexes, Discrete Math., 6 (1973), 377-390.

Column k=3 of A301924.

Cf. A000665 (unlabeled 3-uniform), A025035, A125791 (labeled 3-uniform), A289837, A301922, A302374 (labeled 3-uniform spanning), A302394, A306017, A319540, A320395, A322451 (unlabeled 3-uniform spanning), A323292-A323299.

Inverse Euler transform of A000665. - Sean A. Irvine, Mar 05 2015

a(7)-a(9) corrected and extended by Sean A. Irvine, Mar 05 2015

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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A299471 Regular triangle where T(n,k) is the number of labeled k-uniform hypergraphs spanning n vertices.

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A309858 Number A(n,k) of k-uniform hypergraphs on n unlabeled nodes; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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Maple

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A322451 Number of unlabeled 3-uniform hypergraphs spanning n vertices.

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

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A051240 Number of 4-uniform hypergraphs on n unlabeled nodes, or equivalently pure 3-dimensional simplicial complexes on n unlabeled nodes.

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A301481 Number of unlabeled uniform hypergraphs spanning n vertices.

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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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