%I A102897
%S A102897 2,4,14,122,4960,2771104,151947502948
%N A102897 Number of ACI algebras (or semilattices) on n generators.
%C A102897 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.
%C A102897 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).
%C A102897 An ACI algebra or semilattice is a system with a single binary, idempotent, 
               commutative and associative operation.
%D A102897 V. B. Alekseev, On the number of intersection semilattices [in Russian], 
               Diskretnaya Mat. 1 (1989), 129-136.
%D A102897 G. Birkhoff, Lattice Theory. American Mathematical Society, Colloquium 
               Publications, Vol. 25, 3rd ed., Providence, RI, 1967.
%D A102897 G. Burosch, J. Demetrovics, G. O. H. Katona, D. J. Kleitman and A. A. 
               Sapozhenko, On the number of closure operations, in Combinatorics, 
               Paul Erdos is Eighty (Volume 1), Keszthely: Bolyai Society Mathematical 
               Studies, 1993, 91-105.
%D A102897 Alfred Horn, Journal of Symbolic Logic 16 (1951), 14-21. [See Lemma 7.]
%D A102897 D. E. Knuth, The Art of Computer Programming, Vol. 4, Section 7.1.1 (in 
               preparation).
%D A102897 E. H. Moore, Introduction to a Form of General Analysis, AMS Colloquium 
               Publication 2 (1910), pp. 53-80.
%D A102897 Maria Paola Bonacina and Nachum Dershowitz, Canonical Inference for Implicational 
               Systems, in Automated Reasoning, Lecture Notes in Computer Science, 
               Volume 5195/2008, Springer-Verlag.
%H A102897 N. Dershowitz, G. S. Huang and M. Harris, <a href="http://lat.inf.tu-dresden.de/
               ~harris/enumhorn.pdf">Draft</a>.
%H A102897 D. E. Knuth, <a href="http://www-cs-faculty.stanford.edu/~knuth/programs.html">
               HORN-COUNT</a>
%F A102897 a(n) = 2*A102896(n) = sum( C(n, k)*A102895, k=0..n), where C(n, k) is 
               the binomial coefficient
%F A102897 Asymptotically, log_2 a(n) ~ binomial(n, floor(n/2)) for all of A102894, 
               A102895, A102896 and this sequence [Alekseev; Burosch et al.]
%e A102897 a(2) = 14: Let the points be labeled a, b. We want the number of collections 
               of subsets of {a, b} which are closed under intersection. 0 subsets: 
               1 way ({}), 1 subset: 4 ways (0; a; b; ab), 2 subsets: 5 ways (0,
               a; 0,b; 0,ab; a,ab; b,ab) [not a,b because their intersection, 0, 
               would be missing], 3 subsets: 3 ways (0,a,b; 0,a,ab; 0,b,ab), 4 subsets: 
               1 way (0,a,b,ab), for a total of 14.
%Y A102897 Cf. A102894, A102895, A102896, A108798, A108799, A108800, A108801.
%Y A102897 For nonempty set systems of the same type, see A121921.
%Y A102897 Sequence in context: A005737 A000609 A102449 this_sequence A001527 A067209 
               A134040
%Y A102897 Adjacent sequences: A102894 A102895 A102896 this_sequence A102898 A102899 
               A102900
%K A102897 nonn,hard
%O A102897 0,1
%A A102897 Mitch Harris (Harris.Mitchell(AT)mgh.harvard.edu), Jan 18 2005
%E A102897 Additional comments from D. E. Knuth, Jul 01, 2005