A319189 Number of uniform regular hypergraphs spanning n vertices.

1, 1, 2, 3, 10, 29, 3780, 5012107

Offset: 0

Author

Gus Wiseman, Dec 17 2018

Keywords

• nonn
• more

Comments

We define a hypergraph to be any finite set of finite nonempty sets. A hypergraph is uniform if all edges have the same size, and regular if all vertices have the same degree. The span of a hypergraph is the union of its edges.

Also the number of 0-1 matrices with n columns, all distinct rows, no zero columns, equal row-sums, and equal column-sums, up to a permutation of the rows.

Examples

			The a(4) = 10 edge-sets:
               {{1,2,3,4}}
              {{1,2},{3,4}}
              {{1,3},{2,4}}
              {{1,4},{2,3}}
            {{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}}
  {{1,2},{1,3},{1,4},{2,3},{2,4},{3,4}}
Inequivalent representatives of the a(4) = 10 matrices:
  [1 1 1 1]
.
  [1 1 0 0] [1 0 1 0] [1 0 0 1]
  [0 0 1 1] [0 1 0 1] [0 1 1 0]
.
  [1 0 0 0] [1 1 0 0] [1 1 0 0] [1 0 1 0] [1 1 1 0]
  [0 1 0 0] [1 0 1 0] [1 0 0 1] [1 0 0 1] [1 1 0 1]
  [0 0 1 0] [0 1 0 1] [0 1 1 0] [0 1 1 0] [1 0 1 1]
  [0 0 0 1] [0 0 1 1] [0 0 1 1] [0 1 0 1] [0 1 1 1]
.
  [1 1 0 0]
  [1 0 1 0]
  [1 0 0 1]
  [0 1 1 0]
  [0 1 0 1]
  [0 0 1 1]
		

Crossrefs

Uniform hypergraphs are counted by A306021. Unlabeled uniform regular multiset partitions are counted by A319056. Regular graphs are A295193. Uniform clutters are A299353.

Cf. A002829, A005176, A007016, A049311, A058891, A101370, A110100, A110101, A321717, A321720.

Programs

Mathematica

Table[Sum[SeriesCoefficient[Product[1+Times@@x/@s,{s,Subsets[Range[n],{m}]}],Sequence@@Table[{x[i],0,k},{i,n}]],{m,0,n},{k,1,Binomial[n,m]}],{n,5}]

Extensions

a(7) from Jinyuan Wang, Jun 20 2020