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A051185
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Number of intersecting families of an n-element set. Also number of n-variable clique Boolean functions.
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52
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2, 6, 40, 1376, 1314816, 912818962432, 291201248266450683035648, 14704022144627161780744368338695925293142507520, 12553242487940503914363982718112298267975272720808010757809032705650591023015520462677475328
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OFFSET
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1,1
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COMMENTS
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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
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REFERENCES
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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.
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LINKS
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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
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EXAMPLE
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a(2) = 6 because we have: {}, {{1}}, {{2}}, {{1, 2}}, {{1}, {1, 2}}, {{2}, {1, 2}}. - Geoffrey Critzer, Aug 16 2013
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MATHEMATICA
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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 *)
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CROSSREFS
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Cf. A036239, A051180-A051184.
Sequence in context: A199574 A320489 A135755 * A118623 A000612 A319633
Adjacent sequences: A051182 A051183 A051184 * A051186 A051187 A051188
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KEYWORD
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hard,nonn,nice
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AUTHOR
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Vladeta Jovovic, Goran Kilibarda
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EXTENSIONS
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a(8)-a(9) by Andries E. Brouwer, Aug 07 2012, Dec 11 2012
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STATUS
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approved
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