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

Author

Joshua Moerman, Apr 20 2020

Keywords

• nonn
• more

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.

Examples

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

Links

• 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.

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.

Formula

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

Extensions

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