A322705 - OEIS

A322705

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

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

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OFFSET

0,4

COMMENTS

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.

LINKS

Table of n, a(n) for n=0..7.

EXAMPLE

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}}

MATHEMATICA

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}]

CROSSREFS

Row sums of A322706.

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

Sequence in context: A111195 A323293 A258868 * A167007 A064006 A003095

Adjacent sequences: A322702 A322703 A322704 * A322706 A322707 A322708

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Dec 23 2018

STATUS

approved