A323294 -id:A323294 - cp's OEIS frontend

A289837 Number of cliques in the n-tetrahedral graph.

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

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

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

cp's OEIS Frontend

A289837 Number of cliques in the n-tetrahedral graph.

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

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Author

Keywords

Examples

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Crossrefs

Programs

Mathematica

PARI

Formula

Extensions