A102897 - OEIS

OFFSET

0,1

COMMENTS

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

REFERENCES

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.

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

G. Burosch, J. Demetrovics, G. O. H. Katona, D. J. Kleitman and A. A. Sapozhenko, On the number of closure operations, in Combinatorics, Paul Erdős is Eighty (Volume 1), Keszthely: Bolyai Society Mathematical Studies, 1993, 91-105.

P. Colomb, A. Irlande and O. Raynaud, Counting of Moore Families for n=7, International Conference on Formal Concept Analysis (2010)

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

D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.1, p. 79.

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

LINKS

Table of n, a(n) for n=0..7.

N. Dershowitz, G. S. Huang and M. Harris, Enumeration Problems Related to Ground Horn Theories, arXiv:cs/0610054v2 [cs.LO], 2006-2008.

D. E. Knuth, HORN-COUNT

FORMULA

a(n) = 2*A102896(n) = Sum_{k=0..n} C(n, k)*A102895(k), 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.]

EXAMPLE

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.

From Gus Wiseman, Aug 03 2019: (Start)

The a(0) = 2 through a(2) = 14 sets of subsets closed under union:

{} {} {}

{{}} {{}} {{}}

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

{{},{1}}

{{},{2}}

{{},{1,2}}

{{1},{1,2}}

{{2},{1,2}}

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

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

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

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

(End)

MATHEMATICA

Table[Length[Select[Subsets[Subsets[Range[n]]], SubsetQ[#, Union@@@Tuples[#, 2]]&]], {n, 0, 3}] (* Gus Wiseman, Aug 03 2019 *)

CROSSREFS

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

Regarding sets of subsets closed under union:

- The case with an edge containing all of the vertices is A102895.

- The case without empty edges is A102896.

- The case with intersection instead of union is (also) A102897.

- The unlabeled version is A193675.

- The case closed under both union and intersection is A306445.

- The BII-numbers of set-systems closed under union are A326875.

- The covering case is A326906.

Cf. A102894, A108798, A108800, A193674, A193675, A326866, A326880, A326900.

Sequence in context: A240973 A102449 A193520 * A305856 A001527 A067209

Adjacent sequences: A102894 A102895 A102896 * A102898 A102899 A102900

KEYWORD

nonn,hard,more

AUTHOR

Mitch Harris, Jan 18 2005

EXTENSIONS

Additional comments from Don Knuth, Jul 01 2005

STATUS

approved