A334253 Number of strict closure operators on a set of n elements which satisfy the T_0 separation axiom.

1, 1, 3, 35, 2039, 1352390, 75945052607, 14087646108883940225

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), 81-101.

B. Venkateswarlu and U. M. Swamy, T_0-Closure Operators and Pre-Orders, Lobachevskii Journal of Mathematics, 39 (2018), 1446-1452.

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 set-systems 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