A007153 Dedekind numbers: number of monotone Boolean functions or antichains of subsets of an n-set containing at least one nonempty set. (Formerly M3551 N1439)

0, 1, 4, 18, 166, 7579, 7828352, 2414682040996, 56130437228687557907786, 286386577668298411128469151667598498812364

Offset

0,3

Comments

Equivalently, the number of elements of the free distributive lattice with n generators.

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

The count of antichains excludes the empty antichain which contains no subsets and the antichain consisting of only the empty set.

The number of continuous functions f : R^n->R with f(x_1,..,x_n) in {x_1,..,x_n}. - Jan Fricke, Feb 12 2004

From Robert FERREOL, Mar 23 2009: (Start)

a(n) is also the number of reduced normal conjunctive forms with n variables without negation.

For example the 18 forms for n=3 are :

a

b

c

a or b

a or c

b or c

a or b or c

a and b

a and c

b and c

a and (b or c)

b and (a or c)

c and (a or b)

(a or b) and (a or c)

(b or a) and (b or c)

(c or a) and (c or b)

a and b and c

(a or b) and (a or c) and (b or c)

(End)

References

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

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.

R. Balbes and P. Dwinger, Distributive Lattices, Univ. Missouri Press, 1974, see p. 97. - N. J. A. Sloane, Aug 15 2010

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.

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

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

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.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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

Links

Table of n, a(n) for n=0..9.

K. S. Brown, Dedekind's problem

J. Berman, Free spectra of 3-element algebras, R. S. Freese and O. C. Garcia, editors, Universal Algebra and Lattice Theory (Puebla, 1982), Lect. Notes Math. Vol. 1004, 1983. (Annotated scanned copy)

R. Church, Numerical analysis of certain free distributive lattices, Duke Math. J., 6 (1940), 732-734.

Jacob North Clark, Stephen Montgomery-Smith, Shapley-like values without symmetry, arXiv:1809.07747 [econ.TH], 2018.

J. D. Farley, Was Gelfand right? The many loves of lattice theory, Notices AMS 69:2 (2022), 190-197.

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

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.

N. M. Rivière, Recursive formulas on free distributive lattices, J. Combin. Theory, 5 (1968), 229-234. - N. J. A. Sloane, Aug 15 2010

N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).

M. Ward, Note on the order of the free distributive lattice, Bull. Amer. Math. Soc., (1946), Abstract 52-5-135 p.423. - N. J. A. Sloane, Aug 15 2010

Eric Weisstein's World of Mathematics, Antichain

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

K. Yamamoto, Logarithmic order of free distributive lattice, Math. Soc. Japan, 6 (1954), 343-353. - N. J. A. Sloane, Aug 15 2010

Index entries for sequences related to Boolean functions

Example

a(2)=4 from the antichains {{1}}, {{2}}, {{1,2}}, {{1},{2}}.

Crossrefs

Equals A000372 - 2 and A014466 - 1.. Cf. A003182.

Sequence in context: A356530 A222766 A302827 * A239839 A367486 A156870

Adjacent sequences: A007150 A007151 A007152 * A007154 A007155 A007156

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

Term a(9) (using A000372) from Joerg Arndt, Apr 07 2023

Status

approved