A322704 -id:A322704 - cp's OEIS frontend

A059441 Triangle T(n,k) (n >= 1, 0 <= k <= n-1) giving number of regular labeled graphs with n nodes and degree k, read by rows.

1, 1, 1, 1, 0, 1, 1, 3, 3, 1, 1, 0, 12, 0, 1, 1, 15, 70, 70, 15, 1, 1, 0, 465, 0, 465, 0, 1, 1, 105, 3507, 19355, 19355, 3507, 105, 1, 1, 0, 30016, 0, 1024380, 0, 30016, 0, 1, 1, 945, 286884, 11180820, 66462606, 66462606, 11180820, 286884, 945, 1

Offset: 1

N. J. A. Sloane, Feb 01 2001

			1;
1,   1;
1,   0,       1;
1,   3,       3,        1;
1,   0,      12,        0,          1;
1,  15,      70,       70,         15,    1;
1,   0,     465,        0,        465,    0,   1;
1, 105,    3507,    19355,      19355, 3507, 105, 1;
1,   0,   30016,        0,    1024380, ...;
1, 945,  286884, 11180820,   66462606, ...;
1,   0, 3026655,        0, 5188453830, ...;

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 279.

Andrew Howroyd, Table of n, a(n) for n = 1..300 (rows 1..24)
Denis S. Krotov, [[2,10],[6,6]]-equitable partitions of the 12-cube, arXiv:2012.00038 [math.CO], 2020.
Brendan D. McKay, Applications of a technique for labeled enumeration, Congress. Numerantium, 40 (1983), 207-221. See page 216.
Wikipedia, Regular graph

Row sums are A295193.
Columns: A123023 (k=1), A001205 (k=2), A002829 (k=3, with alternating zeros), A005815 (k=4), A338978 (k=5, with alternating zeros), A339847 (k=6).
Cf. A051031 (unlabeled case), A324163 (connected case), A333351 (multigraphs).
Cf. A001147, A058891, A319189, A319190, A319612, A319729, A322635, A322659, A322698, A322704.

Table[SeriesCoefficient[Product[1+Times@@x/@s,{s,Subsets[Range[n],{2}]}],Sequence@@Table[{x[i],0,k},{i,n}]],{n,9},{k,0,n-1}] (* Gus Wiseman, Dec 24 2018 *)

for(n=1, 10, print(A059441(n))) \\ See A295193 for script, Andrew Howroyd, Aug 28 2019

a(37)-a(55) from Andrew Howroyd, Aug 25 2017

A322705 Number of k-uniform k-regular hypergraphs spanning n vertices, for some 1 <= k <= n.

Original entry on oeis.org

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

Offset: 0

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.

			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}}

Row sums of A322706.

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

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}]

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

Original entry on oeis.org

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

			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}]

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A059441 Triangle T(n,k) (n >= 1, 0 <= k <= n-1) giving number of regular labeled graphs with n nodes and degree k, read by rows.

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A322705 Number of k-uniform k-regular hypergraphs spanning n vertices, for some 1 <= k <= n.

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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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