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A014466 Dedekind numbers: monotone Boolean functions, or nonempty antichains of subsets of an n-set 3
1, 2, 5, 19, 167, 7580, 7828353, 2414682040997, 56130437228687557907787 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

A monotone Boolean function is an increasing functions from P(S), the set of subsets of S, to {0,1}.

The count of antichains includes the antichain consisting of only the empty set, but excludes the empty antichain.

Also counts bases of hereditary systems.

REFERENCES

I. Anderson, Combinatorics of Finite Sets. Oxford Univ. Press, 1987, p. 38.

Arocha, Jorge Luis (1987) "Antichains in ordered sets" [ In Spanish ]. Anales del Instituto de Matematicas de la Universidad Nacional Autonoma de Mexico 27: 1-21.

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.

G. Birkhoff, Lattice Theory. American Mathematical Society, Colloquium Publications, Vol. 25, 3rd ed., Providence, RI, 1967, p. 63.

Donald E. Campbell, Jack Graver and Jerry S. Kelly, There are more strategy-proof procedures than you think, Mathematical Social Sciences 64 (2012) 263-265. - From N. J. A. Sloane, Oct 23 2012

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 273.

J. Dezert, Fondations pour une nouvelle theorie du raisonnement plausible et paradoxal (la DSmT), Tech. Rep. 1/06769 DTIM, ONERA, Paris, page 33, January 2003.

J. Dezert, F. Smarandache, On the generating of hyper-powersets for the DSmT, Proceedings of the 6th International Conference on Information Fusion, Cairns, Australia, 2003.

M. A. Harrison, Introduction to Switching and Automata Theory. McGraw Hill, NY, 1965, p. 188.

D. J. Kleitman, On Dedekind's problem: The number of monotone Boolean functions. Proc. Amer. Math. Soc. 21 1969 677-682.

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.

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.

S. Muroga, Threshold Logic and Its Applications. Wiley, NY, 1971, p. 38 and 214.

D. B. West, Introduction to Graph Theory, 2nd ed., Prentice-Hall, NJ, 2001, p. 349.

D. H. Wiedemann, A computation of the eighth Dedekind number, Order 8 (1991) 5-6.

LINKS

Table of n, a(n) for n=0..8.

K. Atanassov, On Some of Smarandache's Problems

K. S. Brown, Dedekind's problem

Jean Dezert, Foundations for a new theory for plausible and paradoxical reasoning, Tech. Rep. DTIM/IED, ONERA, Paris, pp. 14-15, 2002.

Jean Dezert, Combination of paradoxical sources of information within the neutrosophic framework, Proceedings of the First International Conference on Neutrosophics (2001).

J. L. King, Brick tiling and monotone Boolean functions

Eric Weisstein's World of Mathematics, Antichain

Index entries for sequences related to Boolean functions

EXAMPLE

a(2)=5 from the antichains {{}}, {{1}}, {{2}}, {{1,2}}, {{1},{2}}.

CROSSREFS

Equals A000372 - 1 = A007153 + 1. Cf. A003182.

Sequence in context: A002786 A039719 A198203 * A108799 A193674 A085871

Adjacent sequences:  A014463 A014464 A014465 * A014467 A014468 A014469

KEYWORD

nonn,hard,nice

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Last term from D. H. Wiedemann, personal communication.

Additional comments from Michael Somos, Jun 10 2002.

STATUS

approved

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Last modified October 31 09:36 EDT 2014. Contains 248861 sequences.