A000665 -id:A000665 - cp's OEIS frontend

A309865 Number T(n,k) of k-uniform hypergraphs on n unlabeled nodes; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

2, 2, 2, 2, 3, 2, 2, 4, 4, 2, 2, 5, 11, 5, 2, 2, 6, 34, 34, 6, 2, 2, 7, 156, 2136, 156, 7, 2, 2, 8, 1044, 7013320, 7013320, 1044, 8, 2, 2, 9, 12346, 1788782616656, 29281354514767168, 1788782616656, 12346, 9, 2

Offset: 0

Alois P. Heinz, Aug 20 2019

T(n,k) is defined for n,k >= 0. The triangle contains only the terms with k<=n. T(n,k) = 1 for k>n.

See A000088 and A000665 for more references.

			Triangle T(n,k) begins:
  2;
  2, 2;
  2, 3,    2;
  2, 4,    4,       2;
  2, 5,   11,       5,       2;
  2, 6,   34,      34,       6,    2;
  2, 7,  156,    2136,     156,    7, 2;
  2, 8, 1044, 7013320, 7013320, 1044, 8, 2;
  ...

Alois P. Heinz, Rows n = 0..14, 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. A309858 (the same as square array).

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)):
T:= proc(n, k) option remember; `if`(k>n-k,
      T(n, n-k), b(n$2, [], k))
    end:
seq(seq(T(n, k), k=0..n), n=0..9);

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

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

A323297 Number of 3-uniform hypergraphs on n labeled vertices where no two edges have exactly one vertex in common.

Original entry on oeis.org

1, 1, 1, 2, 16, 76, 271, 1212, 10158, 78290, 503231, 3495966, 33016534, 327625520, 3000119669, 28185006956, 308636238516, 3631959615948, 42031903439809, 493129893459310, 6264992355842706, 84639308481270656, 1159506969481515271, 16131054826385628592

Offset: 0

Gus Wiseman, Jan 11 2019

nonn

			The a(4) = 16 hypergraphs:
  {}
  {{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 8 unlabeled 3-uniform hypergraphs on 6 vertices with no two edges having exactly one vertex 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 {{1,3,4},{2,3,4}}
  10 X {{1,2,3},{4,5,6}}
  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}}

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

Cf. A000665, A025035, A125791, A289837, A299471, A302374, A302394, A306021, A323293, A323294, A323296.

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}]

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

Binomial transform of A323296.

E.g.f.: exp(x - x^2/2 - x^3/3 + 5*x^4/24 + x^2*exp(x)/2). - Andrew Howroyd, Aug 18 2019

a(10)-a(11) from Alois P. Heinz, Aug 11 2019

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

A051249 Pure 4-dimensional simplicial complexes on n unlabeled nodes.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 7, 1044, 1788782616656, 234431745534048922731115555415680, 1994324729203114587259985605157804740271034553359179870979936357974016

Offset: 0

Vladeta Jovovic

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..15

Cf. A000088, A000665, A051240.

Column k=5 of A309858.

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

A319876 Irregular triangle read by rows where T(n,k) is the number of permutations of {1,...,n} whose action on 2-element subsets of {1,...,n} has k cycles.

Original entry on oeis.org

1, 0, 2, 0, 2, 3, 1, 0, 0, 14, 0, 9, 0, 1, 0, 0, 24, 50, 20, 0, 15, 10, 0, 0, 1, 0, 0, 0, 264, 0, 340, 0, 40, 0, 60, 0, 15, 0, 0, 0, 1, 0, 0, 0, 720, 1764, 504, 0, 1120, 630, 0, 0, 70, 105, 105, 0, 0, 21, 0, 0, 0, 0, 1, 0, 0, 0, 0, 13488, 0, 14112, 0, 3724, 0

Offset: 1

Gus Wiseman, Dec 12 2018

The permutation
  1 -> 1
  2 -> 2
  3 -> 4
  4 -> 3
acts on unordered pairs of distinct elements of {1,2,3,4} to give
  (1,2) -> (1,2)
  (1,3) -> (1,4)
  (1,4) -> (1,3)
  (2,3) -> (2,4)
  (2,4) -> (2,3)
  (3,4) -> (3,4)
which has 4 cycles
       (1,2)
  (1,3) <-> (1,4)
  (2,3) <-> (2,4)
       (3,4)
so is counted under T(4,4).

			Triangle begins:
   1
   0   2
   0   2   3   1
   0   0  14   0   9   0   1
   0   0  24  50  20   0  15  10   0   0   1
   0   0   0 264   0 340   0  40   0  60   0  15   0   0   0   1
The T(4,4) = 9 permutations: (1243), (1324), (1432), (2134), (2143), (3214), (3412), (4231), (4321).

Row n has A000124(n - 1) terms. Row sums are the factorial numbers A000142.

Cf. A000088, A000612, A000665, A000666, A003180, A050535, A070166, A317794, A317795.

Table[Length[Select[Permutations[Range[n]],PermutationCycles[Ordering[Map[Sort,Subsets[Range[n],{2}]/.Rule@@@Table[{i,#[[i]]},{i,n}],{1}]],Length]==k&]],{n,5},{k,0,n*(n-1)/2}]

A000088(n) = (1/n!) * Sum_k 2^k * T(n,k).

A092337 Triangle read by rows: T(n,m) = number of 3-uniform hypergraphs with m hyperedges on n unlabeled nodes, where 0 <= m <= C(n,3).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 6, 6, 6, 4, 2, 1, 1, 1, 1, 3, 7, 21, 43, 94, 161, 249, 312, 352, 312, 249, 161, 94, 43, 21, 7, 3, 1, 1, 1, 1, 3, 10, 38, 137, 509, 1760, 5557, 15709, 39433, 87659, 172933, 303277, 473827, 660950, 824410, 920446, 920446, 824410, 660950

Offset: 3

Gordon F. Royle, Mar 18 2004

A 3-uniform hypergraph is a set of 3-subsets of the nodes with isomorphism determined by permutations of the node set. The numbers are calculated using Polya enumeration from the cycle index of the symmetric group Sym(n) in its action on 3-subsets of an n-set, which can easily be computed by MAGMA or GAP. A000665 is the sum of each row of the triangle.

			Triangle T(n,m) begins:
1, 1;
1, 1, 1, 1,  1;
1, 1, 2, 4,  6,  6,  6,   4,   2,   1,   1;
1, 1, 3, 7, 21, 43, 94, 161, 249, 312, 352, 312, 249, 161, 94, 43, 21, 7, 3, 1, 1;

Edgar M. Palmer, On the number of n-plexes, Discrete Math., 6 (1973), 377-390.

Cf. A000665, A051240.

Needs["Combinatorica`"]; Table[A = Subsets[Range[n], {3}];
  CoefficientList[CycleIndex[Replace[Map[Sort,System`PermutationReplace[A, SymmetricGroup[n]], {2}],Table[A[[i]] -> i, {i, 1, Length[A]}], 2], s] /.
Table[s[i] -> 1 + x^i, {i, 1, Binomial[n, 3]}], x], {n,3,7}] // Grid (* Geoffrey Critzer, Oct 28 2015 *)

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

A057783 Building block is 2 hexagons side-by-side; sequence gives number of pieces (polydohexes) that can be formed from n such pairs of hexagons.

Original entry on oeis.org

1, 6, 74, 1257, 25379, 544108, 12037738

Offset: 1

N. J. A. Sloane, Oct 29 2000

Computed by Brendan Owen.

Andrew Clarke, Other Polyforms
Sean A. Irvine, Java program (github)

Cf. A000228, A000105, A000665.

Link updated by William Rex Marshall, Dec 16 2009

a(7) from Sean A. Irvine, Jul 02 2022

cp's OEIS Frontend

A309865 Number T(n,k) of k-uniform hypergraphs on n unlabeled nodes; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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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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Mathematica

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

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A323297 Number of 3-uniform hypergraphs on n labeled vertices where no two edges have exactly one vertex in common.

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A051249 Pure 4-dimensional simplicial complexes on n unlabeled nodes.

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A319876 Irregular triangle read by rows where T(n,k) is the number of permutations of {1,...,n} whose action on 2-element subsets of {1,...,n} has k cycles.

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A092337 Triangle read by rows: T(n,m) = number of 3-uniform hypergraphs with m hyperedges on n unlabeled nodes, where 0 <= m <= C(n,3).

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A003190 Number of connected 2-plexes.