A299353 -id:A299353 - cp's OEIS frontend

A322705 Number of k-uniform k-regular hypergraphs spanning n vertices, for some 1 <= k <= n.

1, 1, 1, 2, 5, 26, 472, 23342

Offset: 0

Gus Wiseman, Dec 23 2018

We define a hypergraph to be any finite set of finite nonempty sets. A hypergraph is k-uniform if all edges contain exactly k vertices, and k-regular if all vertices belong to exactly k edges. The span of a hypergraph is the union of its edges.

			The a(3) = 2 hypergraphs:
  {{1},{2},{3}}
  {{1,2},{1,3},{2,3}}
The a(4) = 5 hypergraphs:
  {{1},{2},{3},{4}}
  {{1,2},{1,3},{2,4},{3,4}}
  {{1,2},{1,4},{2,3},{3,4}}
  {{1,3},{1,4},{2,3},{2,4}}
  {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}
The a(5) = 26 hypergraphs:
  {{1},{2},{3},{4},{5}}
  {{1,2},{1,3},{2,4},{3,5},{4,5}}
  {{1,2},{1,3},{2,5},{3,4},{4,5}}
  {{1,2},{1,4},{2,3},{3,5},{4,5}}
  {{1,2},{1,4},{2,5},{3,4},{3,5}}
  {{1,2},{1,5},{2,3},{3,4},{4,5}}
  {{1,2},{1,5},{2,4},{3,4},{3,5}}
  {{1,3},{1,4},{2,3},{2,5},{4,5}}
  {{1,3},{1,4},{2,4},{2,5},{3,5}}
  {{1,3},{1,5},{2,3},{2,4},{4,5}}
  {{1,3},{1,5},{2,4},{2,5},{3,4}}
  {{1,4},{1,5},{2,3},{2,4},{3,5}}
  {{1,4},{1,5},{2,3},{2,5},{3,4}}
  {{1,2,3},{1,2,4},{1,3,5},{2,4,5},{3,4,5}}
  {{1,2,3},{1,2,4},{1,4,5},{2,3,5},{3,4,5}}
  {{1,2,3},{1,2,5},{1,3,4},{2,4,5},{3,4,5}}
  {{1,2,3},{1,2,5},{1,4,5},{2,3,4},{3,4,5}}
  {{1,2,3},{1,3,4},{1,4,5},{2,3,5},{2,4,5}}
  {{1,2,3},{1,3,5},{1,4,5},{2,3,4},{2,4,5}}
  {{1,2,4},{1,2,5},{1,3,4},{2,3,5},{3,4,5}}
  {{1,2,4},{1,2,5},{1,3,5},{2,3,4},{3,4,5}}
  {{1,2,4},{1,3,4},{1,3,5},{2,3,5},{2,4,5}}
  {{1,2,4},{1,3,5},{1,4,5},{2,3,4},{2,3,5}}
  {{1,2,5},{1,3,4},{1,3,5},{2,3,4},{2,4,5}}
  {{1,2,5},{1,3,4},{1,4,5},{2,3,4},{2,3,5}}
  {{1,2,3,4},{1,2,3,5},{1,2,4,5},{1,3,4,5},{2,3,4,5}}

Row sums of A322706.

Cf. A005176, A058891, A059441, A116539, A295193, A299353, A306021, A319056, A319189, A319190, A319612, A321721, A322704.

Table[Sum[SeriesCoefficient[Product[1+Times@@x/@s,{s,Subsets[Range[n],{k}]}],Sequence@@Table[{x[i],0,k},{i,n}]],{k,1,n}],{n,1,6}]

A302129 Number of unlabeled uniform connected hypergraphs of weight n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 6, 1, 9, 10, 17, 1, 108, 1, 86, 401, 482, 1, 4469, 1, 8435, 47959, 8082, 1, 1007342, 52414, 112835, 15338453, 11899367, 1, 362657533, 1, 977129970, 9349593479, 35787684, 1771297657, 390347162497, 1, 779945988, 9360467497257, 16838238535445

Offset: 0

Gus Wiseman, Jun 20 2018

nonn

A hypergraph is uniform if all edges have the same size. The weight of a hypergraph is the sum of cardinalities of the edges. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(8) = 9 uniform connected hypergraphs:
  {{1,2,3,4,5,6,7,8}}
  {{1,2,3,7}, {4,5,6,7}}
  {{1,2,5,6}, {3,4,5,6}}
  {{1,3,4,5}, {2,3,4,5}}
  {{1,2}, {1,3}, {2,4}, {3,4}}
  {{1,3}, {2,4}, {3,5}, {4,5}}
  {{1,4}, {2,3}, {2,4}, {3,4}}
  {{1,4}, {2,5}, {3,5}, {4,5}}
  {{1,5}, {2,5}, {3,5}, {4,5}}

Andrew Howroyd, Table of n, a(n) for n = 0..50

Cf. A000005, A007716, A038041, A038041, A048143, A299353, A301481, A301920, A301924, A306019, A306021, A331508.

\\ See A331508 for T(n, k).
InvEulerT(v)={my(p=log(1+x*Ser(v))); dirdiv(vector(#v,n,polcoef(p,n)), vector(#v,n,1/n))}
a(n) = {if(n==0, 1, sumdiv(n, d, if(d==1 || d==n, d==1, InvEulerT(vector(d, i, T(n/d, i)))[d] )))} \\ Andrew Howroyd, Jan 16 2024

a(p) = 1 for prime p. - Andrew Howroyd, Jan 16 2024

a(11) onwards from Andrew Howroyd, Jan 16 2024

A321134 Number of uniform hypergraphs spanning n vertices where every two vertices appear together in some edge.

Original entry on oeis.org

1, 1, 1, 2, 7, 406, 505635

Offset: 0

Gus Wiseman, Jan 10 2019

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

			The a(4) = 7 hypergraphs:
  {{1,2,3,4}}
  {{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}}
  {{1,2},{1,3},{1,4},{2,3},{2,4},{3,4}}

Row sums of A320606.

Cf. A058891, A299353, A302394, A306021, A319189, A320444.

Table[Sum[Length[Select[Subsets[Subsets[Range[n],{k}]],And[Union@@#==Range[n],Length[Union@@(Subsets[#,{2}]&/@#)]==Binomial[n,2]]&]],{k,1,n}],{n,1,6}]

cp's OEIS Frontend

A322705 Number of k-uniform k-regular hypergraphs spanning n vertices, for some 1 <= k <= n.

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A302129 Number of unlabeled uniform connected hypergraphs of weight n.

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A321134 Number of uniform hypergraphs spanning n vertices where every two vertices appear together in some edge.

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Mathematica