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

0,2

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).

Links

Table of n, a(n) for n=0..6.

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

Example

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},{}}.

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.

Sequence in context: A061655 A009830 A053374 * A227050 A093876 A322334

Adjacent sequences: A355514 A355515 A355516 * A355518 A355519 A355520

Keyword

nonn,hard,more

Author

Dmitry I. Ignatov, Jul 05 2022

Status

approved