A193674 -id:A193674 - cp's OEIS frontend

A072444 Number of subsets S of the power set P{1,2,...,n} such that: {1}, {2},..., {n} are all elements of S; if X and Y are elements of S and X and Y have a nonempty intersection, then the union of X and Y is an element of S. The sets S are counted modulo permutations on the elements 1,2,...,n.

1, 1, 2, 6, 47, 3095, 26897732

Offset: 0

Wim van Dam (vandam(AT)cs.berkeley.edu), Jun 18 2002

From Gus Wiseman, Aug 01 2019: (Start)
If we define a connectedness system to be a set of finite nonempty sets (edges) that is closed under taking the union of any two overlapping edges, then a(n) is the number of unlabeled connectedness systems on n vertices without singleton edges. Non-isomorphic representatives of the a(3) = 6 connectedness systems without singletons are:
  {}
  {{1,2}}
  {{1,2,3}}
  {{2,3},{1,2,3}}
  {{1,3},{2,3},{1,2,3}}
  {{1,2},{1,3},{2,3},{1,2,3}}
(End)

			a(3) = 6 because of the 6 sets: {{1}, {2}, {3}}; {{1}, {2}, {3}, {1, 2}}; {{1}, {2}, {3}, {1, 2, 3}}; {{1}, {2}, {3}, {1, 2}, {1, 2, 3}}; {{1}, {2}, {3}, {1, 2}, {1, 3}, {1, 2, 3}}; {{1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}.

Wim van Dam, Sub Power Set Sequences

The connected case is A072445.
The labeled case is A072446.
Unlabeled set-systems closed under union are A193674.
Unlabeled connectedness systems are A326867.
Cf. A072447, A092918, A326866, A326871, A326873.

Euler transform of A072445. - Andrew Howroyd, Oct 28 2023

a(0)=1 prepended and a(6) corrected by Andrew Howroyd, Oct 28 2023

A326883 Number of unlabeled set-systems with {} that are closed under intersection and cover n vertices.

Original entry on oeis.org

1, 1, 4, 22, 302, 28630, 216533404, 5592325966377736

Offset: 0

Gus Wiseman, Jul 30 2019

			Non-isomorphic representatives of the a(0) = 1 through a(3) = 22 set-systems:
  {{}}  {{}{1}}  {{}{12}}        {{}{123}}
                 {{}{1}{2}}      {{}{1}{23}}
                 {{}{2}{12}}     {{}{3}{123}}
                 {{}{1}{2}{12}}  {{}{1}{2}{3}}
                                 {{}{23}{123}}
                                 {{}{1}{3}{23}}
                                 {{}{2}{3}{123}}
                                 {{}{3}{13}{23}}
                                 {{}{1}{23}{123}}
                                 {{}{3}{23}{123}}
                                 {{}{1}{2}{3}{23}}
                                 {{}{1}{2}{3}{123}}
                                 {{}{2}{3}{13}{23}}
                                 {{}{1}{3}{23}{123}}
                                 {{}{2}{3}{23}{123}}
                                 {{}{3}{13}{23}{123}}
                                 {{}{1}{2}{3}{13}{23}}
                                 {{}{1}{2}{3}{23}{123}}
                                 {{}{2}{3}{13}{23}{123}}
                                 {{}{1}{2}{3}{12}{13}{23}}
                                 {{}{1}{2}{3}{13}{23}{123}}
                                 {{}{1}{2}{3}{12}{13}{23}{123}}

The case also closed under union is A001930.
The connected case (i.e., with maximum) is A108798.
The same for union instead of intersection is (also) A108798.
The non-covering case is A108800.
The labeled case is A326881.
Cf. A000798, A102894, A102895, A102897, A193674, A193675, A306445, A326875, A326878, A326880.

a(n) = A108800(n) - A108800(n-1) for n > 0. - Andrew Howroyd, Aug 10 2019

a(5)-a(7) from Andrew Howroyd, Aug 10 2019

A326871 Number of unlabeled connectedness systems covering n vertices.

Original entry on oeis.org

1, 1, 4, 24, 436, 80460, 1689114556

Offset: 0

Gus Wiseman, Jul 29 2019

We define a connectedness system (investigated by Vim van Dam in 2002) to be a set of finite nonempty sets (edges) that is closed under taking the union of any two overlapping edges. It is covering if every vertex belongs to some edge.

			Non-isomorphic representatives of the a(0) = 1 through a(3) = 24 connectedness systems:
  {}  {{1}}  {{1,2}}          {{1,2,3}}
             {{1},{2}}        {{1},{2,3}}
             {{2},{1,2}}      {{1},{2},{3}}
             {{1},{2},{1,2}}  {{3},{1,2,3}}
                              {{1},{3},{2,3}}
                              {{2,3},{1,2,3}}
                              {{2},{3},{1,2,3}}
                              {{1},{2,3},{1,2,3}}
                              {{1},{2},{3},{2,3}}
                              {{3},{2,3},{1,2,3}}
                              {{1},{2},{3},{1,2,3}}
                              {{1,3},{2,3},{1,2,3}}
                              {{1},{3},{2,3},{1,2,3}}
                              {{2},{3},{2,3},{1,2,3}}
                              {{2},{1,3},{2,3},{1,2,3}}
                              {{3},{1,3},{2,3},{1,2,3}}
                              {{1,2},{1,3},{2,3},{1,2,3}}
                              {{1},{2},{3},{2,3},{1,2,3}}
                              {{1},{2},{1,3},{2,3},{1,2,3}}
                              {{2},{3},{1,3},{2,3},{1,2,3}}
                              {{3},{1,2},{1,3},{2,3},{1,2,3}}
                              {{1},{2},{3},{1,3},{2,3},{1,2,3}}
                              {{2},{3},{1,2},{1,3},{2,3},{1,2,3}}
                              {{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}

The non-covering case without singletons is A072444.
The case without singletons is A326899.
First differences of A326867 (the non-covering case).
Euler transform of A326869 (the connected case).
The labeled case is A326870.
Cf. A072445, A072446, A072447, A193674, A323818, A326866, A326868.

a(5) from Andrew Howroyd, Aug 10 2019

a(6) from Andrew Howroyd, Oct 28 2023

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).

A326907 Number of non-isomorphic sets of subsets of {1..n} that are closed under union and cover all n vertices. First differences of A193675.

Original entry on oeis.org

2, 2, 6, 28, 330, 28960, 216562364, 5592326182940100

Offset: 0

Gus Wiseman, Aug 03 2019

Differs from A108800 in having a(0) = 2 instead of 1.

			Non-isomorphic representatives of the a(0) = 2 through a(3) = 28 sets of sets:
  {}    {{1}}    {{12}}          {{123}}
  {{}}  {{}{1}}  {{}{12}}        {{}{123}}
                 {{2}{12}}       {{3}{123}}
                 {{}{2}{12}}     {{23}{123}}
                 {{1}{2}{12}}    {{}{3}{123}}
                 {{}{1}{2}{12}}  {{}{23}{123}}
                                 {{1}{23}{123}}
                                 {{3}{23}{123}}
                                 {{13}{23}{123}}
                                 {{}{1}{23}{123}}
                                 {{}{3}{23}{123}}
                                 {{}{13}{23}{123}}
                                 {{2}{3}{23}{123}}
                                 {{2}{13}{23}{123}}
                                 {{3}{13}{23}{123}}
                                 {{12}{13}{23}{123}}
                                 {{}{2}{3}{23}{123}}
                                 {{}{2}{13}{23}{123}}
                                 {{}{3}{13}{23}{123}}
                                 {{}{12}{13}{23}{123}}
                                 {{2}{3}{13}{23}{123}}
                                 {{3}{12}{13}{23}{123}}
                                 {{}{2}{3}{13}{23}{123}}
                                 {{}{3}{12}{13}{23}{123}}
                                 {{2}{3}{12}{13}{23}{123}}
                                 {{}{2}{3}{12}{13}{23}{123}}
                                 {{1}{2}{3}{12}{13}{23}{123}}
                                 {{}{1}{2}{3}{12}{13}{23}{123}}

The case without empty sets is A108798.
The case with a single covering edge is A108800.
First differences of A193675.
The case also closed under intersection is A326898 for n > 0.
The labeled version is A326906.
The same for union instead of intersection is (also) A326907.
Cf. A001930, A102895, A108798, A193674, A193675, A326880, A326881, A326883, A326898, A326908.

a(7) added from A108800 by 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

A326899 Number of unlabeled connectedness systems covering n vertices without singletons.

Original entry on oeis.org

1, 0, 1, 4, 41, 3048, 26894637

Offset: 0

Gus Wiseman, Aug 02 2019

We define a connectedness system (investigated by Vim van Dam in 2002) to be a set of finite nonempty sets (edges) that is closed under taking the union of any two overlapping edges.

			Non-isomorphic representatives of the a(3) = 4 connectedness systems:
  {{1,2,3}}
  {{2,3},{1,2,3}}
  {{1,3},{2,3},{1,2,3}}
  {{1,2},{1,3},{2,3},{1,2,3}}

The case with singletons is A326871.
First differences of A072444 (the non-covering case).
Euler transform of A072445 (the connected case).
The labeled version is A326877.
Cf. A072446, A072447, A193674, A323818, A326866, A326867, A326868, A326869, A326870.

a(6) corrected by Andrew Howroyd, Oct 28 2023

A355517 Number of nonisomorphic systems enumerated by A334254; that is, the number of inequivalent closure operators on a set of n elements where all singletons are closed.

Original entry on oeis.org

1, 2, 1, 4, 50, 7443, 95239971

Offset: 0

Dmitry I. Ignatov, Jul 05 2022

The T_1 axiom states that all singleton sets {x} are closed.

For n>1, this property implies strictness (meaning that the empty set is closed).

			a(0) = 1 counts the empty set, while a(1) = 2 counts {{1}} and {{},{1}}.
For a(2) = 1 the closure system is as follows:  {{1,2},{1},{2},{}}.
The a(3) = 4 inequivalent set-systems of closed sets are:
  {{1,2,3},{1},{2},{3},{}}
  {{1,2,3},{1,2},{1},{2},{3},{}}
  {{1,2,3},{1,2},{1,3},{1},{2},{3},{}}
  {{1,2,3},{1,2},{1,3},{2,3},{1},{2},{3},{}}.

Dmitry I. Ignatov, Supporting iPython code for counting nonequivalent closure systems w.r.t. the T_1 separation axiom, Github repository
Eric Weisstein's World of Mathematics, Separation Axioms
Wikipedia, Separation Axiom

The number of all closure operators is given in A102896, while A193674 contains the number of all nonisomorphic ones.
For T_1 closure operators and their strict counterparts, see A334254 and A334255, respectively; the only difference is a(1).
Cf. A326960, A326961, A326979.

A381472 Number of unlabeled set systems on n vertices which are closed under union of disjoint sets.

Original entry on oeis.org

1, 2, 5, 22, 345, 152589

Offset: 0

Andrew Howroyd, Mar 02 2025

A set system is a finite set of finite nonempty sets.

a(n) is also the number of non-isomorphic sets of subsets of an n-set which are closed under union of disjoint sets and which include the empty set.

			Non-isomorphic representatives of the a(2) = 5 set-systems:
  {}
  {{1}}
  {{1,2}}
  {{1},{1,2}}
  {{1},{2},{1,2}}
The a(3) = 22 non-isomorphic set-systems include A193674(3) = 19 set-systems that are closed under union and 3 additional set-systems which do not include {1,2,3}:
  {{1,2},{1,3}}
  {{1},{1,2},{1,3}}
  {{1,2},{1,3},{2,3}}

Cf. A000612 (set-systems), A193674 (closed under union of sets), A381471, A381575 (labeled case).

A299116 The number of sparse union-closed sets. That is, the number of union-closed sets on n elements containing the empty set and the universe, such that in average each set (not counting the empty set) has at most n/2 elements.

Original entry on oeis.org

0, 0, 0, 2, 27, 3133, 5777931

Offset: 1

Gunnar Brinkmann, Feb 05 2018

If there is a counterexample to the union-closed set conjecture, it is a sparse union-closed set.

G. Brinkmann and R. Deklerck, Generation of Union-Closed Sets and Moore Families, arXiv:1701.03751 [math.CO], 2017.
G. Brinkmann and R. Deklerck, Generation of Union-Closed Sets and Moore Families, Journal of Integer Sequences, Vol.21 (2018), Article 18.1.7.

Cf. A108798, A102894, A193674, A102896.

cp's OEIS Frontend

A072444 Number of subsets S of the power set P{1,2,...,n} such that: {1}, {2},..., {n} are all elements of S; if X and Y are elements of S and X and Y have a nonempty intersection, then the union of X and Y is an element of S. The sets S are counted modulo permutations on the elements 1,2,...,n.

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A326883 Number of unlabeled set-systems with {} that are closed under intersection and cover n vertices.

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A326871 Number of unlabeled connectedness systems covering n vertices.

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

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A326907 Number of non-isomorphic sets of subsets of {1..n} that are closed under union and cover all n vertices. First differences of A193675.

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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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A326899 Number of unlabeled connectedness systems covering n vertices without singletons.

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A355517 Number of nonisomorphic systems enumerated by A334254; that is, the number of inequivalent closure operators on a set of n elements where all singletons are closed.

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A381472 Number of unlabeled set systems on n vertices which are closed under union of disjoint sets.

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A299116 The number of sparse union-closed sets. That is, the number of union-closed sets on n elements containing the empty set and the universe, such that in average each set (not counting the empty set) has at most n/2 elements.

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