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)

Also counts Horn functions on n variables, Boolean functions whose set of truth assignments are closed under 'and', or equivalently, the Boolean functions that can be written as a conjunction of Horn clauses, clauses with at most one negative literal.

Also, number of families of subsets of {1,...,n} that are closed under intersection (because we can throw in the universe, or take it out, without affecting anything else).

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

Also the number of finite sets of finite subsets of {1..n} that are closed under union. - Gus Wiseman, Aug 03 2019

We define a connectedness system (investigated by Vim van Dam in 2002) to be a set of finite nonempty sets (edges) that is closed under taking the union of any two overlapping edges.

From Gus Wiseman, Jul 31 2019: (Start)

If we define a connectedness system to be a set of finite nonempty sets (edges) that is closed under taking the union of any two overlapping edges, then a(n) is the number of connectedness systems on n vertices without singleton edges. The BII-numbers of these set-systems are given by A326873. The a(3) = 12 connectedness systems without singletons are:

{}

{{1,2}}

{{1,3}}

{{2,3}}

{{1,2,3}}

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

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

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

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

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

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

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

(End)

Previous name was: a(1) = 1; for n > 1, a(n) = number of families of subsets of {1, ..., n} that contain both the universe and the empty set, are closed under union of nondisjoint sets, and contain no singletons.

A connectedness system is (as below) a set of (finite) nonempty sets that is closed under union of nondisjoint sets.

The old definition was: "Number of subsets S of the power set P{1,2,...,n} such that: {1}, {2},..., {n} are all elements of S; {1,2,...n} is an element of S; if X and Y are elements of S and X and Y have a nonempty intersection, then the union of X and Y is an element of S."

Comments on the old definition from Gus Wiseman, Aug 01 2019: (Start)

If this sequence were defined similarly to A326877, we would have a(1) = 0.

We define a connectedness system to be a set of finite nonempty sets (edges) that is closed under taking the union of any two overlapping edges. It is connected if it is empty or contains an edge with all the vertices. a(n) is the number of connected connectedness systems on n vertices without singletons. For example, the a(3) = 8 connected connectedness systems without singletons are:

{{1,2,3}}

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

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

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

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

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

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

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

(End)

Conjecture concerning the original definition: a(n) is also the number of families of subsets of {1, ..., n} that contain both the universe and the empty set, are closed under intersection and contain no sets of cardinality n-1. - Tian Vlasic, Nov 04 2022. [This was false, as pointed out by Christian Sievers, Oct 20 2023. It is easy to see that for n>1, a(n) is also the number of families of subsets of {1, ..., n} that contain both the universe and the empty set, are closed under union of nondisjoint sets, and contain no singletons; whereas by duality, the sequence suggested in the conjecture is also the number of those families that are also closed under arbitrary union. For details see the Sievers link. - N. J. A. Sloane, Oct 21 2023]

We define a connectedness system (investigated by Vim van Dam in 2002) to be a set of finite nonempty sets (edges) that is closed under taking the union of any two overlapping edges.

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. Elements of a set-system are sometimes called edges.

The enumeration of these set-systems by number of covered vertices is given by A326870.

We define a connectedness system (investigated by Vim van Dam in 2002) to be a set of finite nonempty sets (edges) that is closed under taking the union of any two overlapping edges. It is covering if every vertex belongs to some edge.