A299471 -id:A299471 - cp's OEIS frontend

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

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

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

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

A302396 Number of families of 4-subsets of an n-set that cover every element.

Original entry on oeis.org

1, 0, 0, 0, 1, 26, 32596, 34359509614, 1180591620442534312297, 85070591730234605240519066638188154620, 1645504557321206042154968331851433202636630333819989444275003856

Offset: 0

Brendan McKay, Apr 07 2018

Number of simple 4-uniform hypergraphs of order n without isolated vertices.

			For n=5 all families with at least two 4-sets will cover every element.

Andrew Howroyd, Table of n, a(n) for n = 0..18

Column 4 of A299471.

Cf. A302394.

Flat(List([0..10],n->Sum([0..n],k->(-1)^k*Binomial(n,k)*2^Binomial(n-k,4)))); # Muniru A Asiru, Apr 07 2018

seq(add((-1)^k * binomial(n,k) * 2^binomial(n-k,4), k = 0..n), n=0..12)

Array[Sum[(-1)^k*Binomial[#, k] 2^Binomial[# - k, 4], {k, 0, #}] &, 11, 0] (* Michael De Vlieger, Apr 07 2018 *)

a(n) = sum(k=0, n, (-1)^k*binomial(n,k)*2^binomial(n-k,4)); \\ Michel Marcus, Apr 07 2018

a(n) = Sum_{k=0..n} (-1)^k * binomial(n,k) * 2^binomial(n-k,4).

A320446 Covers of triangles by tetrahedra: number of labeled 4-uniform hypergraphs spanning n vertices such that every three vertices appear together in some edge.

Original entry on oeis.org

1, 1, 1, 0, 1, 6, 5789

Offset: 0

Gus Wiseman, Jan 10 2019

			The a(5) = 6 hypergraphs:
  {{1234},{1235},{1245},{1345}}
  {{1234},{1235},{1245},{2345}}
  {{1234},{1235},{1345},{2345}}
  {{1234},{1245},{1345},{2345}}
  {{1235},{1245},{1345},{2345}}
  {{1234},{1235},{1245},{1345},{2345}}

Cf. A006126, A038041, A299471, A301922, A302374, A302394, A306021, A319540, A320395, A322451.

Table[Length[Select[Subsets[Subsets[Range[n],{4}]],Length[Union@@(Subsets[#,{3}]&/@#)]==Binomial[n,3]&]],{n,6}]

cp's OEIS Frontend

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

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

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

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PARI

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A302396 Number of families of 4-subsets of an n-set that cover every element.

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A320446 Covers of triangles by tetrahedra: number of labeled 4-uniform hypergraphs spanning n vertices such that every three vertices appear together in some edge.

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