A051185 Number of intersecting families of an n-element set. Also number of n-variable clique Boolean functions.

2, 6, 40, 1376, 1314816, 912818962432, 291201248266450683035648, 14704022144627161780744368338695925293142507520, 12553242487940503914363982718112298267975272720808010757809032705650591023015520462677475328

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

Table of n, a(n) for n=1..9.

Grant Pogosyan, Miyakawa Masahiro, Akihiro Nozaki, Number of Clique Boolean Functions, 1988.

Index entries for sequences related to Boolean functions

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

Cf. A036239, A051180-A051184.

Sequence in context: A320489 A135755 A337699 * A118623 A000612 A319633

Adjacent sequences: A051182 A051183 A051184 * A051186 A051187 A051188

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