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A334254 Number of closure operators on a set of n elements which satisfy the T_1 separation axiom. 5
1, 2, 1, 8, 545, 702525, 66960965307 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
The T_1 axiom states that all singleton sets {x} are closed.
For n>1, this property implies strictness (meaning that the empty set is closed).
LINKS
S. Mapes, Finite atomic lattices and resolutions of monomial ideals, J. Algebra, 379 (2013), 259-276.
Eric Weisstein's World of Mathematics, Separation Axioms
Wikipedia, Separation Axiom
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},{}}
CROSSREFS
The number of all closure operators is given in A102896.
For T_0 closure operators, see A334252.
For strict T_1 closure operators, see A334255, the only difference is a(1).
Sequence in context: A013327 A359625 A009349 * A230582 A011186 A078088
KEYWORD
nonn,more,hard
AUTHOR
Joshua Moerman, Apr 20 2020
EXTENSIONS
a(6) from Dmitry I. Ignatov, Jul 03 2022
STATUS
approved

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Last modified May 29 07:06 EDT 2024. Contains 372926 sequences. (Running on oeis4.)