|
|
A367770
|
|
Number of sets of nonempty non-singleton subsets of {1..n} satisfying a strict version of the axiom of choice.
|
|
18
|
|
|
|
OFFSET
|
0,3
|
|
COMMENTS
|
The axiom of choice says that, given any set of nonempty sets Y, it is possible to choose a set containing an element from each. The strict version requires this set to have the same cardinality as Y, meaning no element is chosen more than once.
Excludes all set-systems with more edges than covered vertices, but this condition is not sufficient.
|
|
LINKS
|
|
|
EXAMPLE
|
The a(3) = 15 set-systems:
{}
{{1,2}}
{{1,3}}
{{2,3}}
{{1,2,3}}
{{1,2},{1,3}}
{{1,2},{2,3}}
{{1,2},{1,2,3}}
{{1,3},{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}}
|
|
MATHEMATICA
|
Table[Length[Select[Subsets[Select[Subsets[Range[n]], Length[#]>1&]], Select[Tuples[#], UnsameQ@@#&]!={}&]], {n, 0, 3}]
|
|
CROSSREFS
|
Set-systems without singletons are counted by A016031, covering A323816.
The complement is counted by A367769.
The complement allowing singletons and empty sets is A367901.
Cf. A059201, A083323, A092918, A102896, A283877, A305000, A306445, A355739, A355740, A367904, A367905.
|
|
KEYWORD
|
nonn,more
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|