A182507 -id:A182507 - cp's OEIS frontend

A326944 Number of T_0 sets of subsets of {1..n} that cover all n vertices, contain {}, and are closed under intersection.

1, 1, 4, 58, 3846, 2685550, 151873991914, 28175291154649937052

Offset: 0

Gus Wiseman, Aug 08 2019

The dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. The T_0 condition means that the dual is strict (no repeated edges).

			The a(0) = 1 through a(2) = 4 sets of subsets:
  {{}}  {{},{1}}  {{},{1},{2}}
                  {{},{1},{1,2}}
                  {{},{2},{1,2}}
                  {{},{1},{2},{1,2}}

The version not closed under intersection is A059201.
The non-T_0 version is A326881.
The version where {} is not necessarily an edge is A326943.
Cf. A003181, A003465, A055621, A182507, A245567, A316978, A319564, A326906, A326939, A326941, A326945, A326947.

dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
Table[Length[Select[Subsets[Subsets[Range[n]]],MemberQ[#,{}]&&Union@@#==Range[n]&&UnsameQ@@dual[#]&&SubsetQ[#,Intersection@@@Tuples[#,2]]&]],{n,0,3}]

a(n) = Sum_{k=0..n} Stirling1(n,k)*A326881(k). - Andrew Howroyd, Aug 14 2019

a(5)-a(7) from Andrew Howroyd, Aug 14 2019

A326945 Number of T_0 sets of subsets of {1..n} that are closed under intersection.

Original entry on oeis.org

2, 4, 12, 96, 4404, 2725942, 151906396568, 28175293281055562650

Offset: 0

Gus Wiseman, Aug 08 2019

The dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. The T_0 condition means that the dual is strict (no repeated edges).

			The a(0) = 2 through a(2) = 12 sets of subsets:
  {}    {}        {}
  {{}}  {{}}      {{}}
        {{1}}     {{1}}
        {{},{1}}  {{2}}
                  {{},{1}}
                  {{},{2}}
                  {{1},{1,2}}
                  {{2},{1,2}}
                  {{},{1},{2}}
                  {{},{1},{1,2}}
                  {{},{2},{1,2}}
                  {{},{1},{2},{1,2}}

The non-T_0 version is A102897.
The version not closed under intersection is A326941.
The covering case is A326943.
The case without empty edges is A326959.
Cf. A003180, A182507, A316978, A319564, A326906, A326939, A326940, A326944, A326947.

Table[Length[Select[Subsets[Subsets[Range[n]]],UnsameQ@@dual[#]&&SubsetQ[#,Intersection@@@Tuples[#,2]]&]],{n,0,3}]

Binomial transform of A326943.

a(5)-a(7) from Andrew Howroyd, Aug 14 2019

A326901 Number of set-systems (without {}) on n vertices that are closed under intersection.

Original entry on oeis.org

1, 2, 6, 32, 418, 23702, 16554476, 1063574497050, 225402367516942398102

Offset: 0

Gus Wiseman, Aug 04 2019

A set-system is a finite set of finite nonempty sets, so no two edges of a set-system that is closed under intersection can be disjoint.

			The a(3) = 32 set-systems:
  {}  {{1}}    {{1}{12}}    {{1}{12}{13}}   {{1}{12}{13}{123}}
      {{2}}    {{1}{13}}    {{2}{12}{23}}   {{2}{12}{23}{123}}
      {{3}}    {{2}{12}}    {{3}{13}{23}}   {{3}{13}{23}{123}}
      {{12}}   {{2}{23}}    {{1}{12}{123}}
      {{13}}   {{3}{13}}    {{1}{13}{123}}
      {{23}}   {{3}{23}}    {{2}{12}{123}}
      {{123}}  {{1}{123}}   {{2}{23}{123}}
               {{2}{123}}   {{3}{13}{123}}
               {{3}{123}}   {{3}{23}{123}}
               {{12}{123}}
               {{13}{123}}
               {{23}{123}}

M. Habib and L. Nourine, The number of Moore families on n = 6, Discrete Math., 294 (2005), 291-296.

The case with union instead of intersection is A102896.
The case closed under union and intersection is A326900.
The covering case is A326902.
The connected case is A326903.
The unlabeled version is A326904.
The BII-numbers of these set-systems are A326905.
Cf. A000798, A001930, A006058, A102895, A102898, A182507, A326866, A326876, A326878, A326882.

Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],SubsetQ[#,Intersection@@@Tuples[#,2]]&]],{n,0,3}]

a(n) = 1 + Sum_{k=0, n-1} binomial(n,k)*A102895(k). - Andrew Howroyd, Aug 10 2019

a(5)-a(8) from Andrew Howroyd, Aug 10 2019

A309615 Number of T_0 set-systems covering n vertices that are closed under intersection.

Original entry on oeis.org

1, 1, 2, 12, 232, 19230, 16113300, 1063117943398, 225402329237199496416

Offset: 0

Gus Wiseman, Aug 11 2019

First differs from A182507 at a(5) = 19230, A182507(5) = 12848.

A set-system is a finite set of finite nonempty sets. The dual of a set-system has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. The T_0 condition means that the dual is strict (no repeated edges).

			The a(0) = 1 through a(3) = 12 set-systems:
  {}  {{1}}  {{1},{1,2}}  {{1},{1,2},{1,3}}
             {{2},{1,2}}  {{2},{1,2},{2,3}}
                          {{3},{1,3},{2,3}}
                          {{1},{1,2},{1,2,3}}
                          {{1},{1,3},{1,2,3}}
                          {{2},{1,2},{1,2,3}}
                          {{2},{2,3},{1,2,3}}
                          {{3},{1,3},{1,2,3}}
                          {{3},{2,3},{1,2,3}}
                          {{1},{1,2},{1,3},{1,2,3}}
                          {{2},{1,2},{2,3},{1,2,3}}
                          {{3},{1,3},{2,3},{1,2,3}}

The version with empty edges allowed is A326943.

Cf. A003465, A059052, A059201, A319637, A326880, A326881, A326901, A326902, A326905, A326944, A326945.

dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Union@@#==Range[n]&&UnsameQ@@dual[#]&&SubsetQ[#,Intersection@@@Tuples[#,2]]&]],{n,0,3}]

a(n) = A326943(n) - A326944(n).
a(n) = Sum_{k = 1..n} s(n,k) * A326901(k - 1) where s = A048994.
a(n) = Sum_{k = 1..n} s(n,k) * A326902(k) where s = A048994.

A326902 Number of set-systems (without {}) covering n vertices that are closed under intersection.

Original entry on oeis.org

1, 1, 3, 19, 319, 21881, 16417973, 1063459099837, 225402359008808647339

Offset: 0

Gus Wiseman, Aug 04 2019

A set-system is a finite set of finite nonempty sets, so no two edges of a set-system that is closed under intersection can be disjoint.

			The a(0) = 1 through a(3) = 19 set-systems:
  {}  {{1}}  {{1,2}}      {{1,2,3}}
             {{1},{1,2}}  {{1},{1,2,3}}
             {{2},{1,2}}  {{2},{1,2,3}}
                          {{3},{1,2,3}}
                          {{1,2},{1,2,3}}
                          {{1,3},{1,2,3}}
                          {{2,3},{1,2,3}}
                          {{1},{1,2},{1,3}}
                          {{2},{1,2},{2,3}}
                          {{3},{1,3},{2,3}}
                          {{1},{1,2},{1,2,3}}
                          {{1},{1,3},{1,2,3}}
                          {{2},{1,2},{1,2,3}}
                          {{2},{2,3},{1,2,3}}
                          {{3},{1,3},{1,2,3}}
                          {{3},{2,3},{1,2,3}}
                          {{1},{1,2},{1,3},{1,2,3}}
                          {{2},{1,2},{2,3},{1,2,3}}
                          {{3},{1,3},{2,3},{1,2,3}}

The case closed under union and intersection is A006058.
The case with union instead of intersection is A102894.
The unlabeled version is A108800(n - 1).
The non-covering case is A326901.
The connected case is A326903.
Cf. A000798, A001930, A102895, A102898, A182507, A326866, A326876, A326878, A326882, A326900, A326901, A326904.

Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Union@@#==Range[n]&&SubsetQ[#,Intersection@@@Tuples[#,2]]&]],{n,0,3}]

Inverse binomial transform of A326901. - Andrew Howroyd, Aug 10 2019

a(5)-a(8) from Andrew Howroyd, Aug 10 2019

A198952 G.f.: Sum_{n>=0} n! * 3^(n(n-1)/2) x^n / Product_{k=1..n} (1 + k3^kx).

Original entry on oeis.org

1, 1, 3, 45, 3267, 991845, 1155605211, 4910640919821, 73614877173054099, 3802910817051064124469, 665332303024345700007225099, 388955052253927480089824057425437, 751710022839628223241451188902204177091

Offset: 0

Paul D. Hanna, May 06 2012

nonn

Compare the g.f. to the identities:
(1) 1/(1-x) = Sum_{n>=0} n! * x^n / Product_{k=1..n} (1 + k*x).
(2) 1+x = Sum_{n>=0} 3^(n*(n-1)/2) * x^n / Product_{k=1..n} (1 + 3^k*x).

			G.f.: A(x) = 1 + x + 3*x^2 + 45*x^3 + 3267*x^4 + 991845*x^5 + 1155605211*x^6 +...
such that
A(x) = 1 + x/(1+3*x) + 2!*3^1*x^2/((1+1*3*x)*(1+2*9*x)) + 3!*3^3*x^3/((1+1*3*x)*(1+2*9*x)*(1+3*27*x)) + 4!*3^6*x^4/((1+1*3*x)*(1+2*9*x)*(1+3*27*x)*(1+4*81*x)) +...

Cf. A182507.

{a(n)=polcoeff(sum(m=0,n,m!*3^(m*(m-1)/2)*x^m/prod(k=1,m,1+k*3^k*x +x*O(x^n))),n)}
for(n=0,20,print1(a(n),", "))

cp's OEIS Frontend

A326944 Number of T_0 sets of subsets of {1..n} that cover all n vertices, contain {}, and are closed under intersection.

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A326945 Number of T_0 sets of subsets of {1..n} that are closed under intersection.

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A326901 Number of set-systems (without {}) on n vertices that are closed under intersection.

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A309615 Number of T_0 set-systems covering n vertices that are closed under intersection.

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A326902 Number of set-systems (without {}) covering n vertices that are closed under intersection.

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A198952 G.f.: Sum_{n>=0} n! * 3^(n*(n-1)/2) * x^n / Product_{k=1..n} (1 + k*3^k*x).

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A198952 G.f.: Sum_{n>=0} n! * 3^(n(n-1)/2) x^n / Product_{k=1..n} (1 + k3^kx).