A326902 -id:A326902 - cp's OEIS frontend

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

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.

A326903 Number of set-systems (without {}) on n vertices that are closed under intersection and have an edge containing all of the vertices, or Moore families without {}.

Original entry on oeis.org

0, 1, 3, 16, 209, 11851, 8277238, 531787248525, 112701183758471199051

Offset: 0

Gus Wiseman, Aug 04 2019

A set-system is a finite set of finite nonempty sets, so no two edges of such a set-system can be disjoint.

If {} is allowed, we get Moore families (A102896, cf A102895).

			The a(1) = 1 through a(3) = 16 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},{1,2,3}}
                      {{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}}

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

The case closed under union and intersection is A006058.
The case with union instead of intersection is A102894.
The unlabeled version is A193674.
The case without requiring the maximum edge is A326901.
The covering case is A326902.
Cf. A000798, A001930, A102895, A102898, A326866, A326876, A326878, A326882, A326904.

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

a(n) = A326901(n) / 2 for n > 0. - Andrew Howroyd, Aug 10 2019

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

A326905 BII-numbers of set-systems (without {}) closed under intersection.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 8, 16, 17, 21, 24, 32, 34, 38, 40, 56, 64, 65, 66, 68, 69, 70, 72, 80, 81, 85, 88, 96, 98, 102, 104, 120, 128, 256, 257, 261, 273, 277, 321, 325, 337, 341, 384, 512, 514, 518, 546, 550, 578, 582, 610, 614, 640, 896, 1024, 1025, 1026, 1028

Offset: 1

Gus Wiseman, Aug 04 2019

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every finite set of finite nonempty sets has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18. Elements of a set-system are sometimes called edges.

			The sequence of all set-systems closed under intersection together with their BII-numbers begins:
   0: {}
   1: {{1}}
   2: {{2}}
   4: {{1,2}}
   5: {{1},{1,2}}
   6: {{2},{1,2}}
   8: {{3}}
  16: {{1,3}}
  17: {{1},{1,3}}
  21: {{1},{1,2},{1,3}}
  24: {{3},{1,3}}
  32: {{2,3}}
  34: {{2},{2,3}}
  38: {{2},{1,2},{2,3}}
  40: {{3},{2,3}}
  56: {{3},{1,3},{2,3}}
  64: {{1,2,3}}
  65: {{1},{1,2,3}}
  66: {{2},{1,2,3}}
  68: {{1,2},{1,2,3}}

The case with union instead of intersection is A326875.
The case closed under union and intersection is A326913.
Set-systems closed under intersection and containing the vertex set are A326903.
Set-systems closed under intersection are A326901, with unlabeled version A326904.
Cf. A006058, A102895, A102898, A326866, A326876, A326878, A326882, A326900, A326902.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[0,100],SubsetQ[bpe/@bpe[#],Intersection@@@Tuples[bpe/@bpe[#],2]]&]

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A326903 Number of set-systems (without {}) on n vertices that are closed under intersection and have an edge containing all of the vertices, or Moore families without {}.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A326905 BII-numbers of set-systems (without {}) closed under intersection.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica