0,3

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)

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a(n) = Sum_{k=0..n} Stirling2(n, k)*A001035(k).

E.g.f.: A(exp(x) - 1) where A(x) is the e.g.f. for A001035. - Geoffrey Critzer, Jul 28 2014

It is known that log_2(a(n)) ~ n^2/4. - Tian Vlasic, Feb 23 2022

From Gus Wiseman, Aug 01 2019: (Start)

The a(3) = 29 topologies are the following (empty sets not shown):

{123} {1}{123} {1}{12}{123} {1}{2}{12}{123} {1}{2}{12}{13}{123}

{2}{123} {1}{13}{123} {1}{3}{13}{123} {1}{2}{12}{23}{123}

{3}{123} {1}{23}{123} {2}{3}{23}{123} {1}{3}{12}{13}{123}

{12}{123} {2}{12}{123} {1}{12}{13}{123} {1}{3}{13}{23}{123}

{13}{123} {2}{13}{123} {2}{12}{23}{123} {2}{3}{12}{23}{123}

{23}{123} {2}{23}{123} {3}{13}{23}{123} {2}{3}{13}{23}{123}

{3}{12}{123}

{3}{13}{123} {1}{2}{3}{12}{13}{23}{123}

{3}{23}{123}

(End)

Table[Length[Select[Subsets[Subsets[Range[n], {1, n}]], Union@@#==Range[n]&&SubsetQ[#, Union[Union@@@Tuples[#, 2], DeleteCases[Intersection@@@Tuples[#, 2], {}]]]&]], {n, 0, 3}] (* Gus Wiseman, Aug 01 2019 *)

Row sums of A326882.

Cf. A001035 (labeled posets), A001930 (unlabeled topologies), A000112 (unlabeled posets), A006057.

Sequences in the Erné (1974) paper: A000798, A001035, A006056, A006057, A001929, A001927, A006058, A006059, A000110.

Cf. A102894, A102895, A102897, A306445, A326866, A326876, A326878, A326881.

For further references concerning the enumeration of topologies and posets see under A001035.

Sequence in context: A231498 A168602 A368452 * A135485 A210526 A221079

Adjacent sequences: A000795 A000796 A000797 * A000799 A000800 A000801

nonn,nice,core,hard

Two more terms from Jobst Heitzig (heitzig(AT)math.uni-hannover.de), Jul 03 2000

a(17)-a(18) are from Brinkmann's and McKay's paper. - Vladeta Jovovic, Jun 10 2007

approved