

A334253


Number of strict closure operators on a set of n elements which satisfy the T_0 separation axiom.


2




OFFSET

0,3


COMMENTS

The T_0 axiom states that the closure of {x} and {y} are different for distinct x and y.
A closure operator is strict if the empty set is closed.


LINKS

Table of n, a(n) for n=0..7.
R. S. R. Myers, J. Adámek, S. Milius, and H. Urbat, Coalgebraic constructions of canonical nondeterministic automata, Theoretical Computer Science, 604 (2015), 81101.
B. Venkateswarlu and U. M. Swamy, T_0Closure Operators and PreOrders, Lobachevskii Journal of Mathematics, 39 (2018), 14461452.


FORMULA

a(n) = Sum_{k=0..n} Stirling1(n,k) * A102894(k).  Andrew Howroyd, Apr 20 2020


EXAMPLE

The a(0) = 1 through a(2) = 3 setsystems of closed sets:
{{}} {{1},{}} {{1,2},{1},{}}
{{1,2},{2},{}}
{{1,2},{1},{2},{}}


CROSSREFS

The number of all strict closure operators is given in A102894.
For all T0 closure operators, see A334252.
For strict T1 closure operators, see A334255.
A strict closure operator which preserves unions is called topological, see A001035.
Cf. A326943, A326944, A326945.
Sequence in context: A320845 A012499 A125530 * A068726 A263512 A093583
Adjacent sequences: A334250 A334251 A334252 * A334254 A334255 A334256


KEYWORD

nonn,more


AUTHOR

Joshua Moerman, Apr 20 2020


EXTENSIONS

a(6)a(7) from Andrew Howroyd, Apr 20 2020


STATUS

approved



