OFFSET
1,1
COMMENTS
Also the number of n-ary Boolean polymorphisms of the binary Boolean relation OR, namely the Boolean functions f(x1,...,xn) with the property that (x1 or y1) and ... and (xn or yn) implies f(x1,...,xn) or f(y1,...,yn). - Don Knuth, Dec 04 2019
These values are necessarily divisible by powers of 2. The sequence of exponents begins 1, 1, 3, 5, 12, 22, 49, 93, ... , 2^(n-1)-C(n-1,floor(n/2)-1), ... (cf. A191391). - Andries E. Brouwer, Aug 07 2012
a(1) = 2^1.
a(2) = 6 = 2^1 * 3
a(3) = 2^3 * 5.
a(4) = 2^5 * 43.
a(5) = 2^12 * 3 * 107.
a(6) = 2^22 * 13 * 16741.
a(7) = 2^49 * 2111 * 245039,
a(8) = 2^93 * 3^2 * 5 * 7211 * 76697 * 59656829,
a(9) = 2^200 * 1823 * 2063 * 576967 * 3600144350906020591.
An intersecting family is a collection of subsets of {1,2,...,n} such that the intersection of every subset with itself or with any other subset in the family is nonempty. The maximum number of subsets in an intersecting family is 2^(n-1). - Geoffrey Critzer, Aug 16 2013
REFERENCES
V. Jovovic, G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, Diskretnaya Matematika, 11 (1999), no. 4, 127-138 (translated in Discrete Mathematics and Applications, 9, (1999), no. 6).
Pogosyan G., Miyakawa M., A. Nozaki, Rosenberg I., The Number of Clique Boolean Functions, IEICE Trans. Fundamentals, Vol. E80-A, No. 8, pp. 1502-1507, 1997/8.
LINKS
Grant Pogosyan, Miyakawa Masahiro, Akihiro Nozaki, Number of Clique Boolean Functions, 1988.
EXAMPLE
a(2) = 6 because we have: {}, {{1}}, {{2}}, {{1, 2}}, {{1}, {1, 2}}, {{2}, {1, 2}}. - Geoffrey Critzer, Aug 16 2013
MATHEMATICA
Table[Length[
Select[Subsets[Subsets[Range[1, n]]],
Apply[And,
Flatten[Table[
Table[Intersection[#[[i]], #[[j]]] != {}, {i, 1,
Length[#]}], {j, 1, Length[#]}]]] &]], {n, 1, 4}] (* Geoffrey Critzer, Aug 16 2013 *)
CROSSREFS
KEYWORD
hard,nonn,nice
AUTHOR
Vladeta Jovovic, Goran Kilibarda
EXTENSIONS
a(8)-a(9) by Andries E. Brouwer, Aug 07 2012, Dec 11 2012
STATUS
approved