A092918 -id:A092918 - 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

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

A323817 Number of connected set-systems covering n vertices with no singletons.

Original entry on oeis.org

1, 0, 1, 12, 1990, 67098648, 144115187673201808, 1329227995784915871895000743748659792, 226156424291633194186662080095093570015284114833799899656335137245499581360

Offset: 0

Gus Wiseman, Jan 30 2019

nonn

			The a(3) = 12 set-systems:
  {{1, 2, 3}}
  {{1, 2}, {1, 3}}
  {{1, 2}, {2, 3}}
  {{1, 3}, {2, 3}}
  {{1, 2}, {1, 2, 3}}
  {{1, 3}, {1, 2, 3}}
  {{2, 3}, {1, 2, 3}}
  {{1, 2}, {1, 3}, {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}}
The A323816(4) - a(4) = 3 disconnected set-systems covering n vertices with no singletons:
  {{1, 2}, {3, 4}}
  {{1, 3}, {2, 4}}
  {{1, 4}, {2, 3}}

G. C. Greubel, Table of n, a(n) for n = 0..11

Cf. A001187, A016031, A048143, A092918, A293510, A317795, A323816 (not necessarily connected), A323818 (with singletons), A323819, A323820 (unlabeled case).

m:=10;
A323816:= func< n | (&+[(-1)^(n-j)*Binomial(n,j)*2^(2^j -j-1): j in [0..n]]) >;
f:= func< x | 1 + Log((&+[A323816(j)*x^j/Factorial(j): j in [0..m+2]])) >;
R:=PowerSeriesRing(Rationals(), m+1);
Coefficients(R!(Laplace( f(x) ))); // G. C. Greubel, Oct 05 2022

b:= n-> add(2^(2^(n-j)-n+j-1)*binomial(n, j)*(-1)^j, j=0..n):
a:= proc(n) option remember; b(n)-`if`(n=0, 0, add(
       k*binomial(n, k)*b(n-k)*a(k), k=1..n-1)/n)
    end:
seq(a(n), n=0..8);  # Alois P. Heinz, Jan 30 2019

nn=10;
ser=Sum[Sum[(-1)^(n-k)*Binomial[n,k]*2^(2^k-k-1),{k,0,n}]*x^n/n!,{n,0,nn}];
Table[SeriesCoefficient[1+Log[ser],{x,0,n}]*n!,{n,0,nn}]

m=10
def A323816(n): return sum((-1)^j*binomial(n,j)*2^(2^(n-j) -n+j-1) for j in range(n+1))
def A323817_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( 1 + log(sum(A323816(j)*x^j/factorial(j) for j in range(m+2))) ).egf_to_ogf().list()
A323817_list(m) # G. C. Greubel, Oct 05 2022

Logarithmic transform of A323816.

A326964 Number of connected set-systems covering a subset of {1..n}.

Original entry on oeis.org

1, 2, 7, 112, 32253, 2147316942, 9223372023968335715, 170141183460469231667123699322514272668, 5789604461865809771178549250434395393752402807429031284280914691514037561273

Offset: 0

Gus Wiseman, Aug 10 2019

nonn

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

			The a(0) = 1 through a(2) = 7 set-systems:
  {}    {}     {}
        {{1}}  {{1}}
               {{2}}
               {{1,2}}
               {{1},{1,2}}
               {{2},{1,2}}
               {{1},{2},{1,2}}

Covering sets of subsets are A000371.
Connected graphs are A001187.
The unlabeled version is A309667.
The BII-numbers of connected set-systems are A326749.
The covering case is A323818.
Cf. A007718, A048143, A058891, A092918, A300913, A304716, A326866, A326948.

csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Length[csm[#]]<=1&]],{n,0,4}]

Binomial transform of A323818.

A309667 Number of non-isomorphic connected set-systems on up to n vertices.

Original entry on oeis.org

1, 2, 5, 35, 1947, 18664537, 12813206150464222, 33758171486592987151274638818642016, 1435913805026242504952006868879460423801146743462225386062178112354069599

Offset: 0

Gus Wiseman, Aug 11 2019

nonn

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

			Non-isomorphic representatives of the a(0) = 1 through a(2) = 5 set-systems:
  {}  {}     {}
      {{1}}  {{1}}
             {{1,2}}
             {{2},{1,2}}
             {{1},{2},{1,2}}

Alois P. Heinz, Table of n, a(n) for n = 0..12

The covering case is A323819 (first differences).
The BII-numbers of connected set-systems are A326749.
The labeled version is A326964.
Cf. A000371, A000612, A001187, A007718, A058891, A092918, A261006, A300913, A323818.

A285616 Triangle read by rows: T(n,k) is the number of hypergraphs on n labeled vertices with exactly k connected components, n>=1, 1<=k<=n.

Original entry on oeis.org

2, 4, 4, 96, 24, 8, 31840, 816, 96, 16, 2147156736, 322240, 4320, 320, 32, 9223372011084915712, 25767883392, 1957440, 18240, 960, 64, 170141183460469231602560095199828453376, 129127208335656968192, 180389362944, 9251200, 67200, 2688, 128

Offset: 1

Geoffrey Critzer, Apr 22 2017

			Triangle begins:
2,
4, 4,
96, 24, 8,
31840, 816, 96, 16,
2147156736, 322240, 4320, 320, 32,
9223372011084915712, 25767883392, 1957440, 18240, 960, 64,
...

Row sums give A058891.

Column 1 is A092918.

nn = 6; A[z_] := Sum[2^(2^n - 1) z^n/n!, {n, 0, nn}];
Map[Select[#, # > 0 &] &,Drop[Range[0, nn]! CoefficientList[
     Series[(A[z]^u), {z, 0, nn}], {z, u}], 1]] // Grid

E.g.f.: A(x)^y where A(x) = Sum_{n>=0} 2^(2^n-1)x^n/n!.

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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A326869 Number of unlabeled connected connectedness systems on n vertices.

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A323817 Number of connected set-systems covering n vertices with no singletons.

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A326964 Number of connected set-systems covering a subset of {1..n}.

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A309667 Number of non-isomorphic connected set-systems on up to n vertices.

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A285616 Triangle read by rows: T(n,k) is the number of hypergraphs on n labeled vertices with exactly k connected components, n>=1, 1<=k<=n.

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