A326903 -id:A326903 - 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

A326904 Number of unlabeled set-systems (without {}) on n vertices that are closed under intersection.

Original entry on oeis.org

1, 2, 4, 10, 38, 368, 29328, 216591692, 5592326399531792

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.

Apart from the offset the same as A193675. - R. J. Mathar, Aug 09 2019

			Non-isomorphic representatives of the a(0) = 1 through a(3) = 10 set-systems:
  {}  {}     {}           {}
      {{1}}  {{1}}        {{1}}
             {{1,2}}      {{1,2}}
             {{2},{1,2}}  {{1,2,3}}
                          {{2},{1,2}}
                          {{3},{1,2,3}}
                          {{2,3},{1,2,3}}
                          {{3},{1,3},{2,3}}
                          {{3},{2,3},{1,2,3}}
                          {{3},{1,3},{2,3},{1,2,3}}

The covering case is A108800(n - 1).
The case with an edge containing all of the vertices is A193674(n - 1).
The case with union instead of intersection is A193674.
The labeled version is A326901.
Cf. A000798, A001930, A006058, A102895, A102898, A326876, A326866, A326878, A326882, A326903, A326906.

a(n > 0) = 2 * A193674(n - 1).

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

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

A326904 Number of unlabeled set-systems (without {}) on n vertices that are closed under intersection.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica