A355517 Number of nonisomorphic systems enumerated by A334254; that is, the number of inequivalent closure operators on a set of n elements where all singletons are closed.

1, 2, 1, 4, 50, 7443, 95239971

Offset: 0

Author

Dmitry I. Ignatov, Jul 05 2022

Keywords

• nonn
• hard
• more

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).

Examples

			a(0) = 1 counts the empty set, while a(1) = 2 counts {{1}} and {{},{1}}.
For a(2) = 1 the closure system is as follows:  {{1,2},{1},{2},{}}.
The a(3) = 4 inequivalent set-systems of closed sets are:
  {{1,2,3},{1},{2},{3},{}}
  {{1,2,3},{1,2},{1},{2},{3},{}}
  {{1,2,3},{1,2},{1,3},{1},{2},{3},{}}
  {{1,2,3},{1,2},{1,3},{2,3},{1},{2},{3},{}}.
		

Links

• Dmitry I. Ignatov, Supporting iPython code for counting nonequivalent closure systems w.r.t. the T_1 separation axiom, Github repository
• Eric Weisstein's World of Mathematics, Separation Axioms
• Wikipedia, Separation Axiom

Crossrefs

The number of all closure operators is given in A102896, while A193674 contains the number of all nonisomorphic ones.

For T_1 closure operators and their strict counterparts, see A334254 and A334255, respectively; the only difference is a(1).

Cf. A326960, A326961, A326979.