A110101 -id:A110101 - 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

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

A110100 a(n) is the number of 2-regular 3-hypergraphs on 3n labeled vertices. (In a 3-hypergraph, each hyper-edge is a proper 3-set; 2-regular implies that each vertex is in exactly 2 hyperedges.)

Original entry on oeis.org

1, 0, 75, 122220, 757275750, 12713292692100, 474415445827323000, 34461884930947363890000, 4431555785100983345799993000, 939388724430508823324694340500000

Offset: 0

Marni Mishna, Jul 11 2005

P-recursive

			One of the 75 2-regular 3-hypergraphs on 6 vertices: {1,2,3} {4,5,6} {1,2,4} {3,5,6}.

Denis S. Krotov, Konstantin V. Vorob'ev, On unbalanced Boolean functions attaining the bound 2n/3-1 on the correlation immunity, arXiv:1812.02166 [math.CO], 2018.
Marni Mishna, Maple worksheet

Cf. A025035, A110101, A110103, A001205.

Recurrence: {a(0) = 1, a(1) = 0, (361631520*n + 1358261784*n^2 + 2841968052*n^3 + 3241507005*n^5 + 3725654130*n^4 + 1922779782*n^6 + 781684101*n^7 + 214347870*n^8 + 37889775*n^9 + 3897234*n^10 + 177147*n^11 + 39916800)*a(n) + (870112800*n + 1655958600*n^2 + 1805971896*n^3 + 561697416*n^5 + 1244162430*n^4 + 166255740*n^6 + 31125384*n^7 + 3346110*n^8 + 157464*n^9 + 199584000)*a(n + 1) + (70976400*n + 86362056*n^2 + 57212568*n^3 + 5161320*n^5 + 22352760*n^4 + 653184*n^6 + 34992*n^7 + 24393600)*a(n + 2) + (-468192*n-411840-198432*n^2-37152*n^3-2592*n^4)*a(n + 3) + 64*a(n + 4), a(2) = 75, a(3) = 122220}.
Differential equation satisfied by generating series A(t)=sum a(n) t^(3n)/(3n)!: {F(0) = 1, 16*t^5*(-2 + t^3)^3*(d^2/dt^2)F(t) + 8*t*(t^9-20*t^3 + 8)*(-2 + t^3)^2*(d/dt)F(t) + t^6*(t^3 + 10)*(t^3-4)*(-2 + t^3)^2*F(t)}.
a(n) ~ 3^(4*n+1/2) * n^(4*n) / (2^n * exp(4*n+1)). - Vaclav Kotesovec, Mar 11 2014

Replaced broken link, Vaclav Kotesovec, Mar 11 2014

A110103 a(n) is the number of 2-regular 4-hypergraphs on 2n labeled vertices. (In a r-hypergraph, each hyper-edge is a proper r-set; k-regular implies that each vertex is in exactly k hyperedges.)

Original entry on oeis.org

1, 0, 0, 15, 1855, 469980, 214402650, 160081596675, 182667234224475, 302414315250247200, 697372026302486234700, 2167773244010692751057625, 8842276105055583472501844625, 46275602006744820263447546152500

Offset: 0

Marni Mishna, Jul 11 2005

P-recursive.

			One of the 15 2-regular 4-hypergraphs on 6 vertices: {{1234},{4561}, {2356}}.

Marni Mishna, Maple worksheet

Cf. A025035, A110100, A110101, A025036.

Differential equation satisfied by exponential generating function sum a(n) t^(2n)/(2n)! {F(0) = 1, -144*t^3*(-2 + t^2)^2*(d^2/dt^2)F(t) - 12*(-2 + t^2)*(2*t^8-t^6 + 72 + 6*t^4-108*t^2)*(d/dt)F(t) - t^5*(-2 + t^2)*(t^2-3)*(t^4 + 4*t^2 + 36)*F(t)}.
Linear recurrence for a(n): {(15067980*n + 10550232*n^6 + 2859384*n^7 + 522720*n^8 + 128*n^11 + 2494800 + 61600*n^9 + 4224*n^10 + 52629038*n^3 + 45995730*n^4 + 26679070*n^5 + 37729494*n^2)*a(n) + (3791790*n + 109368*n^6 + 13872*n^7 + 1008*n^8 + 1247400 + 32*n^9 + 3747208*n^3 + 1767087*n^4 + 543858*n^5 + 4994577*n^2)*a(n + 1) + (28354500*n + 154560*n^6 + 11712*n^7 + 384*n^8 + 15478428*n^3 + 5309976*n^4 + 1152480*n^5 + 27874680*n^2 + 12474000)*a(n + 2) + (-623700-794025*n-48*n^6-115380*n^3-17760*n^4-1440*n^5-416757*n^2)*a(n + 3) + (599130*n + 534600 + 267282*n^2 + 59328*n^3 + 6552*n^4 + 288*n^5)*a(n + 4) + (-14166*n-26730-2484*n^2-144*n^3)*a(n + 5) + 54*a(n + 6), a(3) = 15, a(4) = 1855, a(5) = 469980, a(0) = 1, a(1) = 0, a(2) = 0}
Recurrence (of order 5): 54*(3*n - 4)*a(n) = 18*(n-1)*(2*n - 1)*(12*n^2 - 16*n + 9)*a(n-1) + 18*(n-2)*(n-1)*(2*n - 3)*(2*n - 1)*(3*n + 1)*a(n-2) + 3*(n-2)*(n-1)*(2*n - 5)*(2*n - 3)*(2*n - 1)*(24*n^2 - 65*n + 24)*a(n-3) + 3*(n-3)*(n-2)*(n-1)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*(3*n + 1)*a(n-4) + 2*(n-4)*(n-3)*(n-2)*(n-1)*(2*n - 9)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*(3*n - 1)*a(n-5). - Vaclav Kotesovec, Mar 11 2014
a(n) ~ 2^(3*n+1) * n^(3*n) / (3^n * exp(3*n+3/2)). - Vaclav Kotesovec, Mar 11 2014

Replaced broken link, Vaclav Kotesovec, Mar 11 2014

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

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}]

cp's OEIS Frontend

A319190 Number of regular hypergraphs spanning n vertices.

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A319189 Number of uniform regular hypergraphs spanning n vertices.

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A110100 a(n) is the number of 2-regular 3-hypergraphs on 3n labeled vertices. (In a 3-hypergraph, each hyper-edge is a proper 3-set; 2-regular implies that each vertex is in exactly 2 hyperedges.)

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A110103 a(n) is the number of 2-regular 4-hypergraphs on 2n labeled vertices. (In a r-hypergraph, each hyper-edge is a proper r-set; k-regular implies that each vertex is in exactly k hyperedges.)

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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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