A323293 -id:A323293 - 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

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

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

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

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

A287232 Number of independent vertex sets and vertex covers in the n-tetrahedral graph.

Original entry on oeis.org

271, 5596, 231577, 21286940, 4392750641, 2100400533176

Offset: 6

Eric W. Weisstein, May 22 2017

Eric Weisstein's World of Mathematics, Independent Vertex Set
Eric Weisstein's World of Mathematics, Johnson Graph
Eric Weisstein's World of Mathematics, Tetrahedral Graph
Eric Weisstein's World of Mathematics, Vertex Cover

Essentially the same as A323293.

a(10) from Eric W. Weisstein, Jun 20 2017

a(11) from Eric W. Weisstein, Oct 12 2023

cp's OEIS Frontend

A289837 Number of cliques in the n-tetrahedral graph.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

Extensions

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

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

Extensions

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A287232 Number of independent vertex sets and vertex covers in the n-tetrahedral graph.

Views

Author

Keywords

Links

Crossrefs

Extensions