A326871 -id:A326871 - cp's OEIS frontend

A326867 Number of unlabeled connectedness systems on n vertices.

1, 2, 6, 30, 466, 80926, 1689195482

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.

			Non-isomorphic representatives of the a(0) = 1 through a(3) = 30 connectedness systems:
  {}  {}     {}               {}
      {{1}}  {{1}}            {{1}}
             {{1,2}}          {{1,2}}
             {{1},{2}}        {{1},{2}}
             {{2},{1,2}}      {{1,2,3}}
             {{1},{2},{1,2}}  {{1},{2,3}}
                              {{2},{1,2}}
                              {{1},{2},{3}}
                              {{3},{1,2,3}}
                              {{1},{2},{1,2}}
                              {{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}}

Gus Wiseman, Every Clutter Is a Tree of Blobs, The Mathematica Journal, Vol. 19, 2017.

The case without singletons is A072444.
The labeled case is A326866.
The connected case is A326869.
Partial sums of A326871 (the covering case).
Cf. A072445, A072446, A072447, A102896, A306445, A326870, A326872.

a(5) from Andrew Howroyd, Aug 10 2019

a(6) from Andrew Howroyd, Oct 28 2023

A326870 Number of connectedness systems covering n vertices.

Original entry on oeis.org

1, 1, 5, 77, 6377, 8097721, 1196051135917

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.

			The a(2) = 5 connectedness systems:
  {{1,2}}
  {{1},{2}}
  {{1},{1,2}}
  {{2},{1,2}}
  {{1},{2},{1,2}}

Gus Wiseman, Every Clutter Is a Tree of Blobs, The Mathematica Journal, Vol. 19, 2017.

Inverse binomial transform of A326866 (the non-covering case).
Exponential transform of A326868 (the connected case).
The unlabeled case is A326871.
The BII-numbers of these set-systems are A326872.
The case without singletons is A326877.
Cf. A072445, A072446, A072447, A102896, A323818, A326867, A326869.

Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Union@@#==Range[n]&&SubsetQ[#,Union@@@Select[Tuples[#,2],Intersection@@#!={}&]]&]],{n,0,4}]

a(6) corrected by Christian Sievers, Oct 28 2023

A072445 Number of subsets S of the power set P{1,2,...,n} such that: {1}, {2},..., {n} are all elements of S; {1,2,...,n} is an element 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.

Original entry on oeis.org

1, 1, 1, 4, 40, 3044, 26894586

Offset: 0

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

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. It is connected if it is empty or contains an edge with all the vertices. Then a(n) is the number of unlabeled connected connectedness systems without singletons on n vertices. - Gus Wiseman, Aug 01 2019

			a(3) = 4 because of the 4 sets: {{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
Gus Wiseman, Non-isomorphic representatives of the a(4) = 40 connected connectedness systems without singletons.

The non-connected case is A072444.
The labeled case is A072447.
The case with singletons is A326869.
Cf. A072446, A092918, A108798, A326866, A326867, A326871, A326879.

Inverse Euler transform of A072444. - Andrew Howroyd, Oct 28 2023

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

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.

Original entry on oeis.org

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

A326869 Number of unlabeled connected connectedness systems on n vertices.

Original entry on oeis.org

1, 1, 3, 20, 406, 79964, 1689032658

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 connected if it contains an edge with all the vertices.

			Non-isomorphic representatives of the a(3) = 20 connected connectedness systems:
  {{1,2,3}}
  {{3},{1,2,3}}
  {{2,3},{1,2,3}}
  {{2},{3},{1,2,3}}
  {{1},{2,3},{1,2,3}}
  {{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 case without singletons is A072445.
Connected set-systems are A092918.
The not necessarily connected case is A326867.
The labeled case is A326868.
Euler transform is A326871 (the covering case).
Cf. A072444, A072446, A072447, A102896, A323818, A326866, A326870, A326879.

a(5) from Andrew Howroyd, Aug 16 2019

a(6) from Andrew Howroyd, Oct 28 2023

A326877 Number of connectedness systems covering n vertices without singletons.

Original entry on oeis.org

1, 0, 1, 8, 381, 252080, 18687541309

Offset: 0

Gus Wiseman, Jul 30 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.

			The a(3) = 8 covering connectedness systems without singletons:
  {{1,2,3}}
  {{1,2},{1,2,3}}
  {{1,3},{1,2,3}}
  {{2,3},{1,2,3}}
  {{1,2},{1,3},{1,2,3}}
  {{1,2},{2,3},{1,2,3}}
  {{1,3},{2,3},{1,2,3}}
  {{1,2},{1,3},{2,3},{1,2,3}}

Gus Wiseman, Every Clutter Is a Tree of Blobs, The Mathematica Journal, Vol. 19, 2017.

Inverse binomial transform of A072446 (the non-covering case).
Exponential transform of A072447 if we assume A072447(1) = 0 (the connected case).
The case with singletons is A326870.
The BII-numbers of these set-systems are A326873.
Cf. A072444, A072445, A102896, A323818, A326866, A326867, A326868, A326871, A326872.

Table[Length[Select[Subsets[Subsets[Range[n],{2,n}]],Union@@#==Range[n]&&SubsetQ[#,Union@@@Select[Tuples[#,2],Intersection@@#!={}&]]&]],{n,0,4}]

a(6) corrected by Christian Sievers, Oct 28 2023

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

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A326867 Number of unlabeled connectedness systems on n vertices.

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A326870 Number of connectedness systems covering n vertices.

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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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A326869 Number of unlabeled connected connectedness systems on n vertices.

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A326877 Number of connectedness systems covering n vertices without singletons.

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

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