login
A334255
Number of strict closure operators on a set of n elements which satisfy the T_1 separation axiom.
9
1, 1, 1, 8, 545, 702525, 66960965307
OFFSET
0,4
COMMENTS
The T_1 axiom states that all singleton sets {x} are closed.
A closure operator is strict if the empty set is closed.
LINKS
EXAMPLE
The a(3) = 8 set-systems of closed 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},{}}
MATHEMATICA
Table[Length[
Select[Subsets[Subsets[Range[n]]],
And[MemberQ[#, {}], MemberQ[#, Range[n]],
SubsetQ[#, Intersection @@@ Tuples[#, 2]],
SubsetQ[#, Map[{#} &, Range[n]]]] &]], {n, 0, 4}] (* Tian Vlasic, Jul 29 2022 *)
CROSSREFS
The number of all strict closure operators is given in A102894.
For all strict T_0 closure operators, see A334253.
For T_1 closure operators, see A334254.
Sequence in context: A200706 A266207 A027536 * A174252 A181682 A248331
KEYWORD
nonn,more
AUTHOR
Joshua Moerman, Apr 24 2020
EXTENSIONS
a(6) from Dmitry I. Ignatov, Jul 03 2022
STATUS
approved