cp's OEIS Frontend

A319189 Number of uniform regular hypergraphs spanning n vertices.

%I A319189 #37 Jun 20 2020 08:29:50
%S A319189 1,1,2,3,10,29,3780,5012107
%N A319189 Number of uniform regular hypergraphs spanning n vertices.
%C A319189 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.
%C A319189 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.
%e A319189 The a(4) = 10 edge-sets:
%e A319189                {{1,2,3,4}}
%e A319189               {{1,2},{3,4}}
%e A319189               {{1,3},{2,4}}
%e A319189               {{1,4},{2,3}}
%e A319189             {{1},{2},{3},{4}}
%e A319189         {{1,2},{1,3},{2,4},{3,4}}
%e A319189         {{1,2},{1,4},{2,3},{3,4}}
%e A319189         {{1,3},{1,4},{2,3},{2,4}}
%e A319189     {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}
%e A319189   {{1,2},{1,3},{1,4},{2,3},{2,4},{3,4}}
%e A319189 Inequivalent representatives of the a(4) = 10 matrices:
%e A319189   [1 1 1 1]
%e A319189 .
%e A319189   [1 1 0 0] [1 0 1 0] [1 0 0 1]
%e A319189   [0 0 1 1] [0 1 0 1] [0 1 1 0]
%e A319189 .
%e A319189   [1 0 0 0] [1 1 0 0] [1 1 0 0] [1 0 1 0] [1 1 1 0]
%e A319189   [0 1 0 0] [1 0 1 0] [1 0 0 1] [1 0 0 1] [1 1 0 1]
%e A319189   [0 0 1 0] [0 1 0 1] [0 1 1 0] [0 1 1 0] [1 0 1 1]
%e A319189   [0 0 0 1] [0 0 1 1] [0 0 1 1] [0 1 0 1] [0 1 1 1]
%e A319189 .
%e A319189   [1 1 0 0]
%e A319189   [1 0 1 0]
%e A319189   [1 0 0 1]
%e A319189   [0 1 1 0]
%e A319189   [0 1 0 1]
%e A319189   [0 0 1 1]
%t A319189 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}]
%Y A319189 Uniform hypergraphs are counted by A306021. Unlabeled uniform regular multiset partitions are counted by A319056. Regular graphs are A295193. Uniform clutters are A299353.
%Y A319189 Cf. A002829, A005176, A007016, A049311, A058891, A101370, A110100, A110101, A321717, A321720.
%K A319189 nonn,more
%O A319189 0,3
%A A319189 _Gus Wiseman_, Dec 17 2018
%E A319189 a(7) from _Jinyuan Wang_, Jun 20 2020