The On-Line Encyclopedia of Integer Sequences (OEIS)

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A319189

Number of uniform regular hypergraphs spanning n vertices.
(history; published version)

#37 by Michel Marcus at Sat Jun 20 08:29:50 EDT 2020

STATUS

reviewed

approved

#36 by Joerg Arndt at Sat Jun 20 07:25:35 EDT 2020

STATUS

proposed

reviewed

#35 by Jinyuan Wang at Sat Jun 20 05:51:30 EDT 2020

STATUS

editing

proposed

#34 by Jinyuan Wang at Sat Jun 20 05:51:22 EDT 2020

DATA

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

EXTENSIONS

a(7) from Jinyuan Wang, Jun 20 2020

STATUS

approved

editing

#33 by Susanna Cuyler at Tue Dec 18 08:13:34 EST 2018

STATUS

proposed

approved

#32 by Gus Wiseman at Mon Dec 17 23:00:31 EST 2018

STATUS

editing

proposed

#31 by Gus Wiseman at Mon Dec 17 18:43:36 EST 2018

CROSSREFS

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

#30 by Gus Wiseman at Mon Dec 17 18:40:57 EST 2018

CROSSREFS

Cf. A0028292, A002829, A005176, A007016, A049311, A058891, A101370, A110100, A110100, A321717, A321720.

#29 by Gus Wiseman at Mon Dec 17 18:38:45 EST 2018

NAME

allocated for Gus WisemanNumber of uniform regular hypergraphs spanning n vertices.

DATA

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

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.

Cf. A0028292, A005176, A007016, A049311, A058891, A101370, A110100, A110100, A321717, A321720.

KEYWORD

allocated

nonn,more

AUTHOR

Gus Wiseman, Dec 17 2018

STATUS

approved

editing

#28 by Gus Wiseman at Mon Dec 17 18:38:45 EST 2018

NAME

allocated for Gus Wiseman

KEYWORD

recycled

allocated