A326904 -id:A326904 - cp's OEIS frontend

A193675 Number of nonisomorphic systems enumerated by A102897; that is, the number of inequivalent Horn functions, under permutation of variables.

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

Offset: 0

Don Knuth, Jul 01 2005

When speaking of inequivalent Boolean functions, three groups of symmetries are typically considered: Complementations only, the Abelian group (2,...,2) of 2^n elements; permutations only, the symmetric group of n! elements; or both complementations and permutations, the octahedral group of 2^n n! elements. In this case only symmetry with respect to the symmetric group is appropriate because complementation affects the property of being a Horn function.

Also the number of non-isomorphic sets of subsets of {1..n} that are closed under union. - Gus Wiseman, Aug 04 2019

			From _Gus Wiseman_, Aug 04 2019: (Start)
Non-isomorphic representatives of the a(0) = 2 through a(2) = 10 sets of sets:
  {}    {}        {}
  {{}}  {{}}      {{}}
        {{1}}     {{1}}
        {{},{1}}  {{1,2}}
                  {{},{1}}
                  {{},{1,2}}
                  {{2},{1,2}}
                  {{},{2},{1,2}}
                  {{1},{2},{1,2}}
                  {{},{1},{2},{1,2}}
(End)

D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.1, p. 79.

P. Colomb, A. Irlande and O. Raynaud, Counting of Moore Families for n=7, International Conference on Formal Concept Analysis (2010).
D. E. Knuth, HORN-COUNT

The covering case is A326907.
The case without {} is A193674.
The labeled version is A102897.
The same with intersection instead of union is also A193675.
The case closed under both union and intersection also is A326908.
Cf. A102894, A102895, A102896, A102897, A108798, A108800, A326867, A326875, A326904.

a(n) = 2 * A193674(n).

a(6) received from Don Knuth, Aug 17 2005
a(6) corrected by Pierre Colomb, Aug 02 2011
a(7) = 2*A193674(7) from Hugo Pfoertner, Jun 18 2018

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

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

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

A193675 Number of nonisomorphic systems enumerated by A102897; that is, the number of inequivalent Horn functions, under permutation of variables.

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A326901 Number of set-systems (without {}) on 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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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 {}.

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

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