login
A319189
Number of uniform regular hypergraphs spanning n vertices.
18
1, 1, 2, 3, 10, 29, 3780, 5012107
OFFSET
0,3
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.
EXAMPLE
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]
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}]
CROSSREFS
Uniform hypergraphs are counted by A306021. Unlabeled uniform regular multiset partitions are counted by A319056. Regular graphs are A295193. Uniform clutters are A299353.
Sequence in context: A338583 A121909 A153909 * A301971 A319671 A338593
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Dec 17 2018
EXTENSIONS
a(7) from Jinyuan Wang, Jun 20 2020
STATUS
approved