0,1

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.

V. B. Alekseev, On the number of intersection semilattices [in Russian], Diskretnaya Mat. 1 (1989), 129-136.

G. Birkhoff, Lattice Theory. American Mathematical Society, Colloquium Publications, Vol. 25, 3rd ed., Providence, RI, 1967.

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.

Alfred Horn, Journal of Symbolic Logic 16 (1951), 14-21. [See Lemma 7.]

D. E. Knuth, The Art of Computer Programming, Vol. 4, Section 7.1.1 (in preparation).

E. H. Moore, Introduction to a Form of General Analysis, AMS Colloquium Publication 2 (1910), pp. 53-80.

Maria Paola Bonacina and Nachum Dershowitz, Canonical Inference for Implicational Systems, in Automated Reasoning, Lecture Notes in Computer Science, Volume 5195/2008, Springer-Verlag.

N. Dershowitz, G. S. Huang and M. Harris, Draft.

D. E. Knuth, HORN-COUNT

a(n) = 2*A102896(n) = sum( C(n, k)*A102895, k=0..n), where C(n, k) is the binomial coefficient

Asymptotically, log_2 a(n) ~ binomial(n, floor(n/2)) for all of A102894, A102895, A102896 and this sequence [Alekseev; Burosch et al.]

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.

Cf. A102894, A102895, A102896, A108798, A108799, A108800, A108801.

For nonempty set systems of the same type, see A121921.

Sequence in context: A005737 A000609 A102449 this_sequence A001527 A067209 A134040

Adjacent sequences: A102894 A102895 A102896 this_sequence A102898 A102899 A102900

nonn,hard

Mitch Harris (Harris.Mitchell(AT)mgh.harvard.edu), Jan 18 2005

Additional comments from D. E. Knuth, Jul 01, 2005