A051185 - OEIS

%I #45 Jan 04 2021 20:00:52

%S 2,6,40,1376,1314816,912818962432,291201248266450683035648,

%T 14704022144627161780744368338695925293142507520,

%U 12553242487940503914363982718112298267975272720808010757809032705650591023015520462677475328

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

%C 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

%C 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

%C a(1) = 2^1.

%C a(2) = 6 = 2^1 * 3

%C a(3) = 2^3 * 5.

%C a(4) = 2^5 * 43.

%C a(5) = 2^12 * 3 * 107.

%C a(6) = 2^22 * 13 * 16741.

%C a(7) = 2^49 * 2111 * 245039,

%C a(8) = 2^93 * 3^2 * 5 * 7211 * 76697 * 59656829,

%C a(9) = 2^200 * 1823 * 2063 * 576967 * 3600144350906020591.

%C 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

%D 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).

%D 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.

%H Grant Pogosyan, Miyakawa Masahiro, Akihiro Nozaki, <a href="http://hdl.handle.net/2433/100660">Number of Clique Boolean Functions</a>, 1988.

%H <a href="/index/Bo#Boolean">Index entries for sequences related to Boolean functions</a>

%e a(2) = 6 because we have: {}, {{1}}, {{2}}, {{1, 2}}, {{1}, {1, 2}}, {{2}, {1, 2}}. - _Geoffrey Critzer_, Aug 16 2013

%t Table[Length[

%t   Select[Subsets[Subsets[Range[1, n]]],

%t    Apply[And,

%t      Flatten[Table[

%t        Table[Intersection[#[[i]], #[[j]]] != {}, {i, 1,

%t Length[#]}], {j, 1, Length[#]}]]] &]], {n, 1, 4}] (* _Geoffrey Critzer_, Aug 16 2013 *)

%Y Cf. A036239, A051180-A051184.

%K hard,nonn,nice

%O 1,1

%A _Vladeta Jovovic_, Goran Kilibarda

%E a(8)-a(9) by _Andries E. Brouwer_, Aug 07 2012, Dec 11 2012