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%I A007153 M3551
%S A007153 0,1,4,18,166,7579,7828352,2414682040996,56130437228687557907786
%N A007153 Dedekind numbers: monotone Boolean functions or antichains of subsets 
               of an n-set containing at least one nonempty set.
%C A007153 A monotone Boolean function is an increasing functions from P(S), the 
               set of subsets of S, to {0,1}.
%C A007153 The count of antichains excludes the empty antichain which contains no 
               subsets and the antichain consisting of only the empty set.
%C A007153 The number of continuous functions f : R^n->R with f(x_1,..,x_n) in {x_1,
               ..,x_n}. - Jan Fricke (fricke(AT)math.uni-siegen.de), Feb 12 2004
%C A007153 Comment from Robert FERREOL (rf(AT)mathcurve.com), Mar 23 2009 (Start): 
               a(n) is also the number of reduced normal conjunctive forms with 
               n variables without negation.
%C A007153 For example the 18 forms for n=3 are :
%C A007153 a
%C A007153 b
%C A007153 c
%C A007153 a or b
%C A007153 a or c
%C A007153 b or c
%C A007153 a or b or c
%C A007153 a and b
%C A007153 a and c
%C A007153 b and c
%C A007153 a and (b or c)
%C A007153 b and (a or c)
%C A007153 c and (a or b)
%C A007153 (a or b) and (a or c)
%C A007153 (b or a) and (b or c)
%C A007153 (c or a) and (c or b)
%C A007153 a and b and c
%C A007153 (a or b) and (a or c) and (b or c)
%C A007153 (End)
%D A007153 I. Anderson, Combinatorics of Finite Sets. Oxford Univ. Press, 1987, 
               p. 38.
%D A007153 J. L. Arocha, (1987) "Antichains in ordered sets" [ In Spanish ]. Anales 
               del Instituto de Matematicas de la Universidad Nacional Autonoma 
               de Mexico 27: 1-21.
%D A007153 J. Berman, ``Free spectra of 3-element algebras,'' in R. S. Freese and 
               O. C. Garcia, editors, Universal Algebra and Lattice Theory (Puebla, 
               1982), Lect. Notes Math. Vol. 1004, 1983.
%D A007153 G. Birkhoff, Lattice Theory. American Mathematical Society, Colloquium 
               Publications, Vol. 25, 3rd ed., Providence, RI, 1967, p. 63.
%D A007153 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 273.
%D A007153 M. A. Harrison, Introduction to Switching and Automata Theory. McGraw 
               Hill, NY, 1965, p. 188.
%D A007153 D. J. Kleitman, On Dedekind's problem: The number of monotone Boolean 
               functions. Proc. Amer. Math. Soc. 21 1969 677-682.
%D A007153 D. J. Kleitman and G. Markowsky, On Dedekind's problem: the number of 
               isotone Boolean functions. II. Trans. Amer. Math. Soc. 213 (1975), 
               373-390.
%D A007153 W. F. Lunnon, The IU function: the size of a free distributive lattice, 
               pp. 173-181 of D. J. A. Welsh, editor, Combinatorial Mathematics 
               and Its Applications. Academic Press, NY, 1971.
%D A007153 S. Muroga, Threshold Logic and Its Applications. Wiley, NY, 1971, p. 
               38 and 214.
%D A007153 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A007153 D. B. West, Introduction to Graph Theory, 2nd ed., Prentice-Hall, NJ, 
               2001, p. 349.
%D A007153 D. H. Wiedemann, A computation of the eighth Dedekind number, Order 8 
               (1991) 5-6.
%H A007153 K. S. Brown, <a href="http://www.mathpages.com/home/kmath030.htm">Dedekind's 
               problem</a>
%H A007153 J. L. King, <a href="http://www.math.ufl.edu/~squash/">Brick tiling and 
               monotone Boolean functions</a>
%H A007153 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/sg.txt">
               My favorite integer sequences</a>, in Sequences and their Applications 
               (Proceedings of SETA '98).
%H A007153 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Antichain.html">Link to a section of The World of Mathematics.</a>
%H A007153 <a href="Sindx_Bo.html#Boolean">Index entries for sequences related to 
               Boolean functions</a>
%e A007153 a(2)=4 from the antichains {{1}}, {{2}}, {{1,2}}, {{1},{2}}.
%Y A007153 Equals A000372 - 2 and A014466 - 1.. Cf. A003182.
%Y A007153 Sequence in context: A060841 A059837 A054759 this_sequence A156870 A145075 
               A058924
%Y A007153 Adjacent sequences: A007150 A007151 A007152 this_sequence A007154 A007155 
               A007156
%K A007153 nonn,hard
%O A007153 0,3
%A A007153 N. J. A. Sloane (njas(AT)research.att.com).
%E A007153 Last term from D. H. Wiedemann, personal communication.
%E A007153 Additional comments from Michael Somos, Jun 10 2002.

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