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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.
11
1, 1, 2, 6, 47, 3095, 26897732
OFFSET
0,3
COMMENTS
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)
FORMULA
Euler transform of A072445. - Andrew Howroyd, Oct 28 2023
EXAMPLE
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}}.
CROSSREFS
The connected case is A072445.
The labeled case is A072446.
Unlabeled set-systems closed under union are A193674.
Unlabeled connectedness systems are A326867.
Sequence in context: A078537 A145502 A274702 * A334807 A052596 A344676
KEYWORD
nonn,more
AUTHOR
Wim van Dam (vandam(AT)cs.berkeley.edu), Jun 18 2002
EXTENSIONS
a(0)=1 prepended and a(6) corrected by Andrew Howroyd, Oct 28 2023
STATUS
approved