In June 2002 I submitted the following four 'Sub Power Set Sequences'
to
Neil Sloane's OnLine Encyclopedia of Integer Sequences
with the Anumbers:
A072444,
A072445,
A072446
and
A072447.
Definitions
Consider the subsets S of the power set P({1,2,..,n}) that are closed
under the following property:
 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.
For example, {{1},{2},{1,2},{1,3},{1,2,3}} is such a subset, but
{{1},{2},{1,2},{1,3}} is not.
We use the convention that all single element sets {j} are elements of
S.
We ask the question: "For a given n, how many sets S are
there?"
We can expect this number to grow doubleexponential with n as
there are potentially 2^(2^n) subsets of P({1,...,n}).
The data below indicates that this is indeed probably the case.
We make two choices when counting the sets S:
 Do we force the maximal subset {1,2,...,n} to be part of S?
 How do we count sets that are permutations of other sets?
That is: are {{1},{2},{3},{1,2}} and {{1},{2},{3},{1,3}}
different sets or not?
By varying these options we get the following four sequences.
Sub Power Sets, modulo permutations (A072444):
1, 2, 6, 47, 3095, 26015236,...
Let a(n) be the 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.
The first 6 values are as follows:
 a(1)=1, by {{1}}
 a(2)=2, by the sets {{1},{2}} and {{1},{2},{1,2}}
 a(3)=6, according to 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}}
 Here you can view the lists that show that
a(4)=47
and a(5)=3095.
(Note the shorthand where {1,2,12} stands for {{1},{2},{1,2}}, et
cetera.)
 The fact that a(6)=26015236 was proven with the values b(1),...,b(6)
below.
Sub Power Sets with maximum subset, modulo permutations
(A072445): 1, 1, 4, 40, 3044, 26012090,...
Let b(n) be the 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.
The first 6 values are as follows:
 b(1)=1, by {{1}}
 b(2)=1, by {{1},{2},{1,2}}
 b(3)=4 according to 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}}
 Here you can view the lists that show that b(4)=40
and b(5)=3044.
(Note the shorthand where {1,2,12} stands for {{1},{2},{1,2}}, et
cetera.)
 The fact that b(6)=26012090 was proven by a brute force search
that took several days on a Pentium II computer.
Sub Power Sets (A072446):
1, 2, 12, 420, 254076, 17199454920,...
Let c(n) be the 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 first 5 values are as follows:
 c(1)=1, by {{1}}
 c(2)=2, by {{1},{2}} and {{1},{2},{1,2}}
 c(3)=12 according to the 12 sets
 {{1},{2},{3}}
 {{1},{2},{3},{1,2}}
 {{1},{2},{3},{1,3}}
 {{1},{2},{3},{2,3}}
 {{1},{2},{3},{1,2,3}}
 {{1},{2},{3},{1,2},{1,2,3}}
 {{1},{2},{3},{1,3},{1,2,3}}
 {{1},{2},{3},{2,3},{1,2,3}}
 {{1},{2},{3},{1,2},{1,3},{1,2,3}}
 {{1},{2},{3},{1,2},{2,3},{1,2,3}}
 {{1},{2},{3},{1,3},{2,3},{1,2,3}}
 {{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}
 Here you can view the list that shows that c(4)=420
(Note the shorthand where {1,2,12} stands for {{1},{2},{1,2}}, et
cetera.)
 c(5)=254076
 The fact that c(6)=17199454920 took a week to compute on a Pentium
II.
Sub Power Sets with maximum subset
(A072447): 1, 1, 8, 378, 252000, 17197930224,...
Let d(n) be the 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 first 5 values are as follows:
 d(1)=1, by {{1}}
 d(2)=1, by {{1},{2},{1,2}}
 d(3)=8 according to the 8 sets
 {{1},{2},{3},{1,2,3}}
 {{1},{2},{3},{1,2},{1,2,3}}
 {{1},{2},{3},{1,3},{1,2,3}}
 {{1},{2},{3},{2,3},{1,2,3}}
 {{1},{2},{3},{1,2},{1,3},{1,2,3}}
 {{1},{2},{3},{1,2},{2,3},{1,2,3}}
 {{1},{2},{3},{1,3},{2,3},{1,2,3}}
 {{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}
 Here you can view the list that shows that
d(4)=378
(Note the shorthand where {1,2,12} stands for {{1},{2},{1,2}}, et
cetera.)
 d(5)=252000
 The determination of the value d(6)=17197930224 was part of the
computation of c(6).
last update: August 2006 by vandam@cs
