cp's OEIS Frontend

From Altug Alkan, Dec 18 2015 and Feb 28 2017: (Start)

a(p^k) == k+1 (mod p) for all primes p. This is proved by Kizmaz at On The Number Of Topologies On A Finite Set link. For proof see Theorem 2.4 in page 2 and 3. So a(19) == 2 (mod 19).

a(p+n) == A265042(n) (mod p) for all primes p. This is also proved by Kizmaz at related link, see Theorem 2.7 in page 4. If n=2 and p=17, a(17+2) == A265042(2) (mod 17), that is a(19) == 51 (mod 17). So a(19) is divisible by 17.

In conclusion, a(19) is a number of the form 323*n - 17. (End)

The BII-numbers of finite topologies without their empty set are given by A326876. - Gus Wiseman, Aug 01 2019

From Tian Vlasic, Feb 23 2022: (Start)

Although no general formula is known for a(n), by considering the number of topologies with a fixed number of open sets, it is possible to explicitly represent the sequence in terms of Stirling numbers of the second kind.

For example: a(n,3) = 2*S(n,2), a(n,4) = S(n,2) + 6*S(n,3), a(n,5) = 6*S(n,3) + 24*S(n,4).

Lower and upper bounds are known: 2^n <= a(n) <= 2^(n*(n-1)), n > 1.

This follows from the fact that there are 2^(n*(n-1)) reflexive relations on a set with n elements.

Furthermore: a(n+1) <= a(n)*(3a(n)+1). (End)

Or, number of Moore families on an n-set, that is, families of subsets that contain the universal set {1,...,n} and are closed under intersection.

Or, number of closure operators on a set of n elements.

An ACI algebra or semilattice is a system with a single binary, idempotent, commutative and associative operation.

Also the number of set-systems on n vertices that are closed under union. The BII-numbers of these set-systems are given by A326875. - Gus Wiseman, Jul 31 2019

From Bernhard Ganter, Jul 08 2025: (Start)

Also the number of union-free families of subsets of an n-set; i.e., families of nonempty sets on n elements such that no set is a union of some others.

Also the number of intersection-free families of subsets of an n-set; i.e., of families of proper subsets on n elements such that no set is an intersection of some others.

(Note that every union-free family on an n-set is the set of union-irreducible elements of exactly one union-closed family, and each family of union-irreducible elements is union-free. Same for intersection.) (End)

An ACI algebra or semilattice is a system with a single binary, idempotent, commutative and associative operation.

Or, number of families of subsets of {1, ..., n} that are closed under intersection and contain the empty set.

A finite topology is a finite set of finite sets closed under union and intersection and containing {} and the vertex set.

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every finite set of finite nonempty sets has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18.

The enumeration of finite topologies by number of points is given by A000798.

From Gus Wiseman, Aug 05 2019: (Start)

For n > 0, also the number of topologies covering {1..n} whose nonempty open sets have nonempty intersection. Also the number of topologies covering {1..n} whose nonempty open sets are pairwise intersecting. For example, the a(0) = 1 through a(3) = 16 topologies (empty sets not shown) are:

{} {{1}} {{1,2}} {{1,2,3}}

{{1},{1,2}} {{1},{1,2,3}}

{{2},{1,2}} {{2},{1,2,3}}

{{3},{1,2,3}}

{{1,2},{1,2,3}}

{{1,3},{1,2,3}}

{{2,3},{1,2,3}}

{{1},{1,2},{1,2,3}}

{{1},{1,3},{1,2,3}}

{{2},{1,2},{1,2,3}}

{{2},{2,3},{1,2,3}}

{{3},{1,3},{1,2,3}}

{{3},{2,3},{1,2,3}}

{{1},{1,2},{1,3},{1,2,3}}

{{2},{1,2},{2,3},{1,2,3}}

{{3},{1,3},{2,3},{1,2,3}}

(End)

Differs from A102895 in having a(0) = 2 instead of 1.