Search: id:A000372 Results 1-1 of 1 results found. %I A000372 M0817 N0309 %S A000372 2,3,6,20,168,7581,7828354,2414682040998,56130437228687557907788 %N A000372 Dedekind numbers: number of monotone Boolean functions of n variables or number of antichains of subsets of an n-set. %C A000372 A monotone Boolean function is an increasing functions from P(S), the set of subsets of S, to {0,1}. %C A000372 The count of antichains includes the empty antichain which contains no subsets and the antichain consisting of only the empty set. %C A000372 a(n) is also equal to the number of upsets of an n-set S. A set U of subsets of S is an upset if whenever A is in U and B is a superset of A then B is in U. - W. Edwin Clark (eclark(AT)math.usf.edu), Nov 06 2003 %C A000372 Also the inverse binomial transform of A006126 with a 2 prepended to the sequence. - Rodrigo A. Obando (R.Obando(AT)computer.org), Jul 26 2004 %D A000372 I. Anderson, Combinatorics of Finite Sets. Oxford Univ. Press, 1987, p. 38. %D A000372 J. L. Arocha, Antichains in ordered sets [in Spanish], Anales del Instituto de Matematicas de la Universidad Nacional Autonoma de Mexico, 27 (1987), 1-21. %D A000372 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 A000372 J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124. %D A000372 G. Birkhoff, Lattice Theory. American Mathematical Society, Colloquium Publications, Vol. 25, 3rd ed., Providence, RI, 1967, p. 63. %D A000372 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 273. %D A000372 E. N. Gilbert, Lattice theoretic properties of frontal switching functions, J. Math. Phys., 33 (1954), 57-67, see Table III. %D A000372 Sylvain Guilley, Laurent Sauvage, Jean-Luc Danger, Tarik Graba and Yves Mathieu, "Evaluation of Power-Constant Dual-Rail Logic as a Protection of Cryptographic Applications in FPGAs", SSIRI - Secure System Integration and Reliability Improvement, Yokohama: Japan (2008), pp 16-23, DOI: 10.1109/SSIRI.2008.31 ; http://hal.archives-ouvertes.fr/hal-00259153/ en/ [From Sylvain GUILLEY (Sylvain.Guilley(AT)TELECOM-ParisTech.fr), Aug 20 2009] %D A000372 M. A. Harrison, Introduction to Switching and Automata Theory. McGraw Hill, NY, 1965, p. 188. %D A000372 J. Kahn, Entropy, independent sets and antichains, Entropy, independent sets and antichains: a new approach to Dedekind's problem, Proc. Amer. Math. Soc. 130 (2002), no. 2, 371-378. %D A000372 D. J. Kleitman, On Dedekind's problem: The number of monotone Boolean functions. Proc. Amer. Math. Soc. 21 1969 677-682. %D A000372 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 A000372 A. D. Korshunov, The number of monotone Boolean functions, Problemy Kibernet. No. 38, (1981), 5-108, 272. MR0640855 (83h:06013) %D A000372 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 A000372 S. Muroga, Threshold Logic and Its Applications. Wiley, NY, 1971, p. 38 and 214. %D A000372 R. A. Obando, On the number of nondegenerate monotone boolean functions of n variables in an n-variable boolean algebra. In preparation. %D A000372 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). %D A000372 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). %D A000372 D. B. West, Introduction to Graph Theory, 2nd ed., Prentice-Hall, NJ, 2001, p. 349. %D A000372 D. H. Wiedemann, A computation of the eighth Dedekind number, Order 8 (1991) 5-6. %H A000372 K. S. Brown, Dedekind's problem %H A000372 K. S. Brown, Asymptotic upper and lower bounds %H A000372 J. L. King, Brick tiling and monotone Boolean functions %H A000372 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics. %H A000372 R. Zeno, A007501 is an upper bound %H A000372 Index entries for sequences related to Boolean functions %H A000372 R. A. Obando, Project: A map of a rule space (to be posted). %F A000372 The asymptotics can be found in the Korshunov paper. - Boris Bukh (brbukh(AT)yahoo.com), Nov 07 2003 %F A000372 a(n) = Sum_{k=1..n}C(n, k)*b(k) + 2, where b(k) is the number of antichain covers of a labeled n-set A006126. E.g. a(3) = 3*1 + 3*2 + 1*9 + 2 = 20 - Rodrigo A. Obando (R.Obando(AT)computer.org), Jul 26 2004 %e A000372 a(2)=6 from the antichains {}, {{}}, {{1}}, {{2}}, {{1,2}}, {{1},{2}}. %Y A000372 Equals A014466 + 1, also A007153 + 2. Cf. A003182, A059119. %Y A000372 Sequence in context: A124066 A093447 A002078 this_sequence A123930 A125601 A025239 %Y A000372 Adjacent sequences: A000369 A000370 A000371 this_sequence A000373 A000374 A000375 %K A000372 nonn,hard,nice %O A000372 0,1 %A A000372 N. J. A. Sloane (njas(AT)research.att.com). %E A000372 Last term from D. H. Wiedemann, personal communication. %E A000372 Additional comments from Michael Somos, Jun 10 2002. Search completed in 0.002 seconds