A059441 -id:A059441 - cp's OEIS frontend

A322706 Regular triangle read by rows where T(n,k) is the number of k-regular k-uniform hypergraphs spanning n vertices.

1, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 12, 12, 1, 0, 1, 70, 330, 70, 1, 0, 1, 465, 11205, 11205, 465, 1, 0, 1, 3507, 505505, 2531200, 505505, 3507, 1, 0

Offset: 1

Gus Wiseman, Dec 23 2018

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.

			Triangle begins:
  1
  1       0
  1       1       0
  1       3       1       0
  1      12      12       1       0
  1      70     330      70       1       0
  1     465   11205   11205     465       1       0
  1    3507  505505 2531200  505505    3507       1       0
Row 4 counts the following hypergraphs:
  {{1}{2}{3}{4}}  {{12}{13}{24}{34}}  {{123}{124}{134}{234}}
                  {{12}{14}{23}{34}}
                  {{13}{14}{23}{24}}

Row sums are A322705. Second column is A001205. Third column is A110101.

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

Table[Table[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}]

A361254 Number of n-regular graphs on 2*n labeled nodes.

Original entry on oeis.org

1, 1, 3, 70, 19355, 66462606, 2977635137862, 1803595358964773088, 15138592322753242235338875, 1793196665025885172290508971592750, 3040059281615704147007085764679679740691838, 74597015246986083384362428357508730776063716190667288, 26737694395324301026230134763403079891362936970900741153038680278

Offset: 0

Atabey Kaygun, Mar 06 2023

nonn

These graphs share the same degree sequence as the complete bipartite graphs K(n,n).

Atabey Kaygun, Counting Graphs with a Prescribed Degree Sequence
Atabey Kaygun, Enumerating Labeled Graphs that Realize a Fixed Degree Sequence, arXiv:2101.02299 [math.CO], 2021.

Cf. A001223, A059441, A339987, A360437.

\\ See Links in A295193 for GraphsByDegreeSeq.
a(n)={if(n==0, 1, vecsum(GraphsByDegreeSeq(2*n, n, (p, r)->valuation(p,x) >= n-r)[, 2])) } \\ Andrew Howroyd, Mar 06 2023

a(n) = A059441(2*n, n).

a(11)-a(12) from Andrew Howroyd, Mar 06 2023

A374842 Number of 7-regular labeled graphs on 2n nodes.

Original entry on oeis.org

1, 0, 0, 0, 1, 286884, 480413921130, 1803595358964773088, 15138592322753242235338875, 271849772205948458085090804526392, 9883018890803233316233360724489799227748, 689121157937951859333538097288863665976145304960

Offset: 0

Marni Mishna, Jul 23 2024

nonn

These entries are generated by a linear recurrence.

			For example, for n=4, a(4)=1 indicates that there is a single 7-regular graph on 2n=8 vertices. Specifically, this is the complete graph.

Marni Mishna, Table of n, a(n) for n = 0..89
Frédéric Chyzak and Marni Mishna, Differential equations satisfied by generating functions of 5-, 6-, and 7-regular labelled graphs: a reduction-based approach, arXiv:2406.04753 [math.CO], 2024.

Alternating terms of column k=7 of A059441.

Cf. A165628 (unlabeled case).

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A322706 Regular triangle read by rows where T(n,k) is the number of k-regular k-uniform hypergraphs spanning n vertices.

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A361254 Number of n-regular graphs on 2*n labeled nodes.

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A374842 Number of 7-regular labeled graphs on 2n nodes.

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