|
|
A326969
|
|
Number of sets of subsets of {1..n} whose dual is a weak antichain.
|
|
8
|
|
|
|
OFFSET
|
0,1
|
|
COMMENTS
|
The dual of a set of subsets has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. A weak antichain is a multiset of sets, none of which is a proper subset of any other.
|
|
LINKS
|
|
|
FORMULA
|
a(n) = 2 * Sum_{k = 0..n} binomial(n, k) * A326970(k).
|
|
EXAMPLE
|
The a(0) = 2 through a(2) = 12 sets of subsets:
{} {} {}
{{}} {{}} {{}}
{{1}} {{1}}
{{},{1}} {{2}}
{{1,2}}
{{},{1}}
{{},{2}}
{{1},{2}}
{{},{1,2}}
{{},{1},{2}}
{{1},{2},{1,2}}
{{},{1},{2},{1,2}}
|
|
MATHEMATICA
|
dual[eds_]:=Table[First/@Position[eds, x], {x, Union@@eds}];
stableQ[u_, Q_]:=!Apply[Or, Outer[#1=!=#2&&Q[#1, #2]&, u, u, 1], {0, 1}];
Table[Length[Select[Subsets[Subsets[Range[n]]], stableQ[dual[#], SubsetQ]&]], {n, 0, 3}]
|
|
CROSSREFS
|
Sets of subsets whose dual is strict are A326941.
The BII-numbers of set-systems whose dual is a weak antichain are A326966.
Sets of subsets whose dual is a (strict) antichain are A326967.
The case without empty edges is A326968.
|
|
KEYWORD
|
nonn,more
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|