A334252 Number of closure operators on a set of n elements which satisfy the T_0 separation axiom.

1, 2, 5, 44, 2179, 1362585, 75953166947, 14087646640499308474

Offset

0,2

Comments

The T_0 axiom states that the closure of {x} and {y} are different for distinct x and y.

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) * A102896(k). - Andrew Howroyd, Apr 20 2020

Example

The a(0) = 1 through a(2) = 5 set-systems of closed sets:
{{}}  {{}}      {{1,2},{1}}
      {{1},{}}  {{1,2},{2}}
                {{1,2},{1},{}}
                {{1,2},{2},{}}
                {{1,2},{1},{2},{}}

Crossrefs

The number of all closure operators is given in A102896.

For strict T0 closure operators, see A334253.

For T1 closure operators, see A334254.

Cf. A326943, A326944, A326945.

Sequence in context: A221682 A366406 A088309 * A307147 A056680 A005166

Adjacent sequences: A334249 A334250 A334251 * A334253 A334254 A334255

Keyword

nonn,more

Author

Joshua Moerman, Apr 20 2020

Extensions

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

Status

approved