A049311 -id:A049311 - cp's OEIS frontend

A319190 Number of regular hypergraphs spanning n vertices.

1, 1, 3, 19, 879, 5280907, 1069418570520767

Offset: 0

Gus Wiseman, Dec 17 2018

We define a hypergraph to be any finite set of finite nonempty sets. A hypergraph is regular if all vertices have the same degree. The span of a hypergraph is the union of its edges.

			The a(3) = 19 regular hypergraphs:
                 {{1,2,3}}
                {{1},{2,3}}
                {{2},{1,3}}
                {{3},{1,2}}
               {{1},{2},{3}}
            {{1},{2,3},{1,2,3}}
            {{2},{1,3},{1,2,3}}
            {{3},{1,2},{1,2,3}}
            {{1,2},{1,3},{2,3}}
           {{1},{2},{3},{1,2,3}}
           {{1},{2},{1,3},{2,3}}
           {{1},{3},{1,2},{2,3}}
           {{2},{3},{1,2},{1,3}}
        {{1,2},{1,3},{2,3},{1,2,3}}
       {{1},{2},{1,3},{2,3},{1,2,3}}
       {{1},{3},{1,2},{2,3},{1,2,3}}
       {{2},{3},{1,2},{1,3},{1,2,3}}
      {{1},{2},{3},{1,2},{1,3},{2,3}}
  {{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}

Column sums of A188445.

Cf. A002829, A005176, A049311, A058891, A110100, A110101, A116539, A283877, A295193, A306017, A319189.

Table[Sum[SeriesCoefficient[Product[1+Times@@x/@s,{s,Subsets[Range[n],{1,n}]}],Sequence@@Table[{x[i],0,k},{i,n}]],{k,1,2^n}],{n,5}]

a(6) from Andrew Howroyd, Mar 12 2020

A319565 Number of non-isomorphic connected strict T_0 multiset partitions of weight n.

Original entry on oeis.org

1, 1, 1, 4, 8, 21, 62, 175, 553, 1775, 6007

Offset: 0

Gus Wiseman, Sep 23 2018

In a multiset partition, two vertices are equivalent if in every block the multiplicity of the first is equal to the multiplicity of the second. The T_0 condition means that there are no equivalent vertices.

The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 8 multiset partitions:
1:      {{1}}
2:     {{1,1}}
3:    {{1,1,1}}
      {{1,2,2}}
     {{1},{1,1}}
     {{2},{1,2}}
4:   {{1,1,1,1}}
     {{1,2,2,2}}
    {{1},{1,1,1}}
    {{1},{1,2,2}}
    {{2},{1,2,2}}
    {{1,2},{2,2}}
    {{1,3},{2,3}}
   {{1},{2},{1,2}}

Cf. A007716, A007718, A049311, A056156, A059201, A283877, A316980.

Cf. A319557, A319558, A319559, A319560, A319564, A319566, A319567.

A054976 Number of binary n X n matrices with no zero rows or columns, up to row and column permutation.

Original entry on oeis.org

1, 3, 17, 179, 3835, 200082, 29610804, 13702979132, 20677458750966, 103609939177198046, 1745061194503344181714, 99860890306900024150675406, 19611238933283757244479826044874, 13340750149227624084760722122669739026, 31706433098827528779057124372265863803044450

Offset: 1

Vladeta Jovovic, May 27 2000

Also the number of non-isomorphic set multipartitions (multisets of sets) with n parts and n vertices. - Gus Wiseman, Nov 18 2018

			From _Gus Wiseman_, Nov 18 2018: (Start)
Inequivalent representatives of the a(3) = 17 matrices:
  100 100 100 100 100 010 010 001 001 001 001 110 101 101 011 011 111
  100 010 001 011 011 001 101 001 101 011 111 101 011 011 011 111 111
  011 001 011 011 111 111 011 111 011 111 111 011 011 111 111 111 111
Non-isomorphic representatives of the a(1) = 1 through a(3) = 17 set multipartitions:
  {{1}}  {{1},{2}}      {{1},{2},{3}}
         {{2},{1,2}}    {{1},{1},{2,3}}
         {{1,2},{1,2}}  {{1},{3},{2,3}}
                        {{1},{2,3},{2,3}}
                        {{2},{1,3},{2,3}}
                        {{2},{3},{1,2,3}}
                        {{3},{1,3},{2,3}}
                        {{3},{3},{1,2,3}}
                        {{1,2},{1,3},{2,3}}
                        {{1},{2,3},{1,2,3}}
                        {{1,3},{2,3},{2,3}}
                        {{3},{2,3},{1,2,3}}
                        {{1,3},{2,3},{1,2,3}}
                        {{2,3},{2,3},{1,2,3}}
                        {{3},{1,2,3},{1,2,3}}
                        {{2,3},{1,2,3},{1,2,3}}
                        {{1,2,3},{1,2,3},{1,2,3}}
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..50

Column sums of A057150.

Cf. A007716, A048291 (labeled), A049311, A057149, A057151, A104601, A104602, A319616, A320808.

A002724 = Cases[Import["https://oeis.org/A002724/b002724.txt", "Table"], {, }][[All, 2]];
A002725 = Cases[Import["https://oeis.org/A002725/b002725.txt", "Table"], {, }][[All, 2]];
a[n_] := A002724[[n + 1]] - 2 A002725[[n]] + A002724[[n]];
a /@ Range[1, 13] (* Jean-François Alcover, Sep 14 2019 *)

a(n) = A002724(n) - 2*A002725(n-1) + A002724(n-1).

More terms from David Wasserman, Mar 06 2002

Terms a(14) and beyond from Andrew Howroyd, Apr 11 2020

A319189 Number of uniform regular hypergraphs spanning n vertices.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Dec 17 2018

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.

			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]

Uniform hypergraphs are counted by A306021. Unlabeled uniform regular multiset partitions are counted by A319056. Regular graphs are A295193. Uniform clutters are A299353.

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

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}]

a(7) from Jinyuan Wang, Jun 20 2020

A319560 Number of non-isomorphic strict T_0 multiset partitions of weight n.

Original entry on oeis.org

1, 1, 2, 6, 15, 40, 121, 353, 1107, 3550, 11818

Offset: 0

Gus Wiseman, Sep 23 2018

In a multiset partition, two vertices are equivalent if in every block the multiplicity of the first is equal to the multiplicity of the second. The T_0 condition means that there are no equivalent vertices.

The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 15 multiset partitions:
1: {{1}}
2: {{1,1}}
   {{1},{2}}
3: {{1,1,1}}
   {{1,2,2}}
   {{1},{1,1}}
   {{1},{2,2}}
   {{2},{1,2}}
   {{1},{2},{3}}
4: {{1,1,1,1}}
   {{1,2,2,2}}
   {{1},{1,1,1}}
   {{1},{1,2,2}}
   {{1},{2,2,2}}
   {{1},{2,3,3}}
   {{2},{1,2,2}}
   {{1,1},{2,2}}
   {{1,2},{2,2}}
   {{1,3},{2,3}}
   {{1},{2},{1,2}}
   {{1},{2},{2,2}}
   {{1},{2},{3,3}}
   {{1},{3},{2,3}}
   {{1},{2},{3},{4}}

Cf. A007716, A007718, A049311, A053419, A056156, A059201, A283877, A316980.

Cf. A319557, A319558, A319559, A319564, A319565, A319566, A319567.

A323795 Number of non-isomorphic weight-n sets of non-overlapping sets of sets.

Original entry on oeis.org

1, 1, 3, 8, 27, 82, 310, 1163

Offset: 0

Gus Wiseman, Jan 28 2019

Also the number of non-isomorphic set partitions of set-systems of weight n.
All sets and multisets must be finite, and only the outermost may be empty.
The weight of an atom is 1, and the weight of a multiset is the sum of weights of its elements, counting multiplicity.

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 27 multiset partitions:
  {{1}}  {{12}}      {{123}}          {{1234}}
         {{1}{2}}    {{1}{12}}        {{1}{123}}
         {{1}}{{2}}  {{1}{23}}        {{12}{13}}
                     {{1}}{{12}}      {{1}{234}}
                     {{1}}{{23}}      {{12}{34}}
                     {{1}{2}{3}}      {{1}}{{123}}
                     {{1}}{{2}{3}}    {{1}{2}{12}}
                     {{1}}{{2}}{{3}}  {{1}{2}{13}}
                                      {{12}}{{13}}
                                      {{1}}{{234}}
                                      {{1}{2}{34}}
                                      {{12}}{{34}}
                                      {{1}}{{2}{12}}
                                      {{12}}{{1}{2}}
                                      {{1}}{{2}{13}}
                                      {{12}}{{1}{3}}
                                      {{1}}{{2}{34}}
                                      {{1}{2}{3}{4}}
                                      {{12}}{{3}{4}}
                                      {{2}}{{1}{13}}
                                      {{1}}{{2}}{{12}}
                                      {{1}}{{2}}{{13}}
                                      {{1}}{{2}}{{34}}
                                      {{1}}{{2}{3}{4}}
                                      {{1}{2}}{{3}{4}}
                                      {{1}}{{2}}{{3}{4}}
                                      {{1}}{{2}}{{3}}{{4}}

Cf. A004111, A007716, A049311, A050326, A050343, A283877, A306186, A318566.

Cf. A323787, A323788, A323789, A323790, A323791, A323792, A323793, A323794.

A319781 Number of multiset partitions of integer partitions of n with empty intersection. Number of relatively prime factorizations of Heinz numbers of integer partitions of n.

Original entry on oeis.org

1, 0, 0, 1, 3, 9, 21, 48, 103, 214, 436, 863, 1689

Offset: 0

Gus Wiseman, Sep 27 2018

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			The a(3) = 1 through a(5) = 9 multiset partitions:
3: {{1},{2}}
4: {{1},{3}}
   {{2},{1,1}}
   {{1},{1},{2}}
5: {{1},{4}}
   {{2},{3}}
   {{3},{1,1}}
   {{1},{2,2}}
   {{1},{1},{3}}
   {{1},{2},{2}}
   {{2},{1,1,1}}
   {{1},{2},{1,1}}
   {{1},{1},{1},{2}}

Cf. A007716, A049311, A281116, A283877, A316980, A317752, A317755, A317757, A318715.

Cf. A319775, A319779, A319778, A319783.

A057151 Number of square binary matrices with n ones, with no zero rows or columns, up to row and column permutation.

Original entry on oeis.org

1, 1, 2, 4, 8, 18, 41, 102, 252, 666, 1789, 5031, 14486, 43280, 132777, 420267, 1366307, 4566966, 15661086, 55081118, 198425478, 731661754, 2758808581, 10629386376, 41814350148, 167830018952, 686822393793, 2864024856054, 12162059027416, 52564545391789

Offset: 1

Vladeta Jovovic, Aug 14 2000

nonn

Number of square binary matrices with n ones and with no zero rows or columns is A104602(n). - Vladeta Jovovic, Mar 25 2006

Also the number of non-isomorphic square set multipartitions (multisets of sets) of weight n. A multiset partition or hypergraph is square if its length (number of blocks or edges) is equal to its number of vertices. The weight of a multiset partition is the sum of sizes of its parts. - Gus Wiseman, Nov 16 2018

			There are 666 square binary matrices with 10 ones, with no zero rows or columns, up to row and column permutation: 33 of size 4 X 4, 248 of size 5 X 5, 288 of size 6 X 6, 79 of size 7 X 7, 15 of size 8 X 8, 2 of size 9 X 9 and 1 of size 10 X 10. Cf. A057150.
From _Gus Wiseman_, Nov 16 2018: (Start)
Non-isomorphic representatives of the a(1) = 1 through a(6) = 18 square set multipartitions:
  {1}  {1}{2}  {2}{12}    {12}{12}      {1}{23}{23}      {12}{13}{23}
               {1}{2}{3}  {1}{1}{23}    {2}{13}{23}      {1}{23}{123}
                          {1}{3}{23}    {2}{3}{123}      {13}{23}{23}
                          {1}{2}{3}{4}  {3}{13}{23}      {3}{23}{123}
                                        {3}{3}{123}      {1}{1}{1}{234}
                                        {1}{2}{2}{34}    {1}{1}{24}{34}
                                        {1}{2}{4}{34}    {1}{1}{4}{234}
                                        {1}{2}{3}{4}{5}  {1}{2}{34}{34}
                                                         {1}{3}{24}{34}
                                                         {1}{3}{4}{234}
                                                         {1}{4}{24}{34}
                                                         {1}{4}{4}{234}
                                                         {2}{4}{12}{34}
                                                         {3}{4}{12}{34}
                                                         {4}{4}{12}{34}
                                                         {1}{2}{3}{3}{45}
                                                         {1}{2}{3}{5}{45}
                                                         {1}{2}{3}{4}{5}{6}
(End)

Max Alekseyev, Table of n, a(n) for n = 1..30

Cf. A049311, A056037, A056079, A056080, A057149, A057150, A057152.

Cf. A054976, A101370, A104601, A104602, A120732, A283877, A319616.

More terms from Max Alekseyev, May 31 2007

A319755 Number of non-isomorphic intersecting set multipartitions (multisets of sets) of weight n.

Original entry on oeis.org

1, 1, 2, 3, 6, 9, 19, 30, 60, 107, 212

Offset: 0

Gus Wiseman, Sep 27 2018

A set multipartition is intersecting if no two parts are disjoint. The weight of a set multipartition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(1) = 1 through a(5) = 9 set multipartitions:
1: {{1}}
2: {{1,2}}
   {{1},{1}}
3: {{1,2,3}}
   {{2},{1,2}}
   {{1},{1},{1}}
4: {{1,2,3,4}}
   {{3},{1,2,3}}
   {{1,2},{1,2}}
   {{1,3},{2,3}}
   {{2},{2},{1,2}}
   {{1},{1},{1},{1}}
5: {{1,2,3,4,5}}
   {{4},{1,2,3,4}}
   {{1,4},{2,3,4}}
   {{2,3},{1,2,3}}
   {{2},{1,2},{1,2}}
   {{3},{3},{1,2,3}}
   {{3},{1,3},{2,3}}
   {{2},{2},{2},{1,2}}
   {{1},{1},{1},{1},{1}}

Cf. A007716, A049311, A283877, A305854, A306006, A316980, A319616.

Cf. A319748, A319752, A319759, A319760, A319765, A319779, A319787, A319789.

A068313 Number of (0,1)-matrices with sum of entries equal to n and no zero rows or columns, with weakly decreasing row sums and column sums.

Original entry on oeis.org

1, 4, 15, 82, 457, 3231, 24055, 209375, 1955288, 20455936, 229830841, 2828166755, 37228913365, 528635368980, 7990596990430, 128909374528433, 2202090635802581, 39837079499488151, 759320365206705013, 15234890522990662422, 320634889654149218205, 7068984425261215971205

Offset: 1

Axel Kohnert (axel.kohnert(AT)uni-bayreuth.de), Feb 25 2002

nonn

This is the sum over the matrix of base change from the elementary symmetric functions to the monomial symmetric functions.

			a(2) = 4 because there are 4 different 0-1 matrices of weight 2: 1 10 01 11,1, 01, 10.
From _Gus Wiseman_, Nov 15 2018: (Start)
The a(3) = 15 matrices:
  [1 1 1]
.
  [1 1] [1 1 0] [1 0 1] [0 1 1]
  [1 0] [0 0 1] [0 1 0] [1 0 0]
.
  [1] [1 0] [1 0] [1 0 0] [1 0 0] [0 1] [0 1 0] [0 1 0] [0 0 1] [0 0 1]
  [1] [1 0] [0 1] [0 1 0] [0 0 1] [1 0] [1 0 0] [0 0 1] [1 0 0] [0 1 0]
  [1] [0 1] [1 0] [0 0 1] [0 1 0] [1 0] [0 0 1] [1 0 0] [0 1 0] [1 0 0]
(End)

I. G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford 1979, p. 57.

Ludovic Schwob, Table of n, a(n) for n = 1..37
Ludovic Schwob, On the enumeration of double cosets and self-inverse double cosets, arXiv:2506.04007 [math.CO], 2025. See p. 14.

Cf. A000219, A001970, A007716, A049311, A101370, A117433, A120733, A321646, A321652, A321653, A321654.

prs2mat[prs_]:=Table[Count[prs,{i,j}],{i,Union[First/@prs]},{j,Union[Last/@prs]}];
Table[Length[Select[Subsets[Tuples[Range[n],2],{n}],And[Union[First/@#]==Range[Max@@First/@#],Union[Last/@#]==Range[Max@@Last/@#],OrderedQ[Total/@prs2mat[#]],OrderedQ[Total/@T[prs2mat[#]]]]&]],{n,5}] (* Gus Wiseman, Nov 15 2018 *)

Name changed by Gus Wiseman, Nov 15 2018

a(20) onwards from Ludovic Schwob, Oct 13 2023

cp's OEIS Frontend

A319190 Number of regular hypergraphs spanning n vertices.

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A319565 Number of non-isomorphic connected strict T_0 multiset partitions of weight n.

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A054976 Number of binary n X n matrices with no zero rows or columns, up to row and column permutation.

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A319189 Number of uniform regular hypergraphs spanning n vertices.

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Mathematica

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A319560 Number of non-isomorphic strict T_0 multiset partitions of weight n.

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A323795 Number of non-isomorphic weight-n sets of non-overlapping sets of sets.

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A319781 Number of multiset partitions of integer partitions of n with empty intersection. Number of relatively prime factorizations of Heinz numbers of integer partitions of n.

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A057151 Number of square binary matrices with n ones, with no zero rows or columns, up to row and column permutation.

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Author

Keywords

Comments

Examples

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Extensions

A319755 Number of non-isomorphic intersecting set multipartitions (multisets of sets) of weight n.

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Author

Keywords

Comments

Examples

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A068313 Number of (0,1)-matrices with sum of entries equal to n and no zero rows or columns, with weakly decreasing row sums and column sums.

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