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A000372 Dedekind numbers or Dedekind's problem: number of monotone Boolean functions of n variables, number of antichains of subsets of an n-set, number of elements in a free distributive lattice on n generators, number of Sperner families.
(Formerly M0817 N0309)
2, 3, 6, 20, 168, 7581, 7828354, 2414682040998, 56130437228687557907788 (list; graph; refs; listen; history; text; internal format)



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 empty antichain which contains no subsets and the antichain consisting of only the empty set.

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, Nov 06 2003

Also the number of simple games with n players in minimal winning form. - Fabián Riquelme, May 29 2011

The unlabeled case is A003182. - Gus Wiseman, Feb 20 2019


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

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.

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.

J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124.

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.

R. Dedekind, Über Zerlegungen von Zahlen durch ihre grössten gemeinsamen Teiler, Festschrift Hoch. Braunschweig u. ges. Werke(II), 1897, pp. 103-148.

E. N. Gilbert, Lattice theoretic properties of frontal switching functions, J. Math. Phys., 33 (1954), 57-67, see Table III.

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

D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.1, p. 79.

A. D. Korshunov, The number of monotone Boolean functions, Problemy Kibernet. No. 38, (1981), 5-108, 272. MR0640855 (83h:06013)

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.

R. A. Obando, On the number of nondegenerate monotone boolean functions of n variables in an n-variable boolean algebra. In preparation.

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.


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

Frank a Campo, Relations between powers of Dedekind Numbers and exponential sums related to them, J. Int. Seq. 21 (2018) 18.4.4.

Valentin Bakoev, Combinatorial and Algorithmic Properties of One Matrix Structure at Monotone Boolean Functions, arXiv:1902.06110 [cs.DM], 2019.

Raymond Balbes, On counting Sperner families, J. Combin. Theory Ser. A 27 (1979), no. 1, 1--9. MR0541338 (81b:05010)

R. Baumann and H. Strass, On the number of bipolar Boolean functions, 2014, preprint.

R. Baumann and H. Strass, On the number of bipolar Boolean functions, Journal of Logic and Computation, 27(8) (2017), 2431-2449.

J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124. [Annotated scanned copy]

S. Bolus, A QOBDD-based Approach to Simple Games, Dissertation, Doktor der Ingenieurwissenschaften der Technischen Fakultät der Christian-Albrechts-Universität zu Kiel, 2012. - N. J. A. Sloane, Dec 22 2012

K. S. Brown, Dedekind's problem.

K. S. Brown, Asymptotic upper and lower bounds.

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. - N. J. A. Sloane, Oct 23 2012

Randolph Church, Numerical analysis of certain free distributive structures, Duke Math. J. 6 (1940). 732--734. MR0002842 (2,120c) [According to Math Reviews, gives a(5) incorrectly as 7579. - N. J. A. Sloane, Mar 19 2012]

R. Church, Numerical analysis of certain free distributive structures, Duke Math. J. 6 (1940). 732--734. [Scanned annotated copy]

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

Ori Davidov and Shyamal Peddada, Order-Restricted Inference for Multivariate Binary Data With Application to Toxicology, Journal of the American Statistical Association, Dec 01 2011, 106(496): 1394-1404, doi:10.1198/jasa.2011.tm10322.

Patrick De Causmaecker and Stefan De Wannemacker, Partitioning in the space of anti-monotonic functions, arXiv:1103.2877 [math.NT], 2011.

Patrick De Causmaecker, Stefan De Wannemacker, On the number of antichains of sets in a finite universe, arXiv:1407.4288 [math.CO], 2014 (see Table 1).

P. De Causmaecker, S. De Wannemacker, J. Yellen, Intervals of Antichains and Their Decompositions, arXiv preprint arXiv:1602.04675 [math.CO], 2016.

Conor Finn, Joseph T. Lizier, Generalised Measures of Multivariate Information Content, arXiv:1909.12166 [cs.IT], 2019.

Christian Gießen, Monotone Functions on Bitstrings - Some Structural Notes, Theory of Randomized Optimization Heuristics, Dagstuhl Seminar 17191 (2017), 3.12, p. 33.

Milton W. Green, Letter to N. J. A. Sloane, 1973.

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

Pieter-Jan Hoedt, Parallelizing with MPI in Java to Find the ninth Dedekind Number, preprint, 2015.

Liviu Ilinca, and Jeff Kahn, Counting maximal antichains and independent sets, arXiv:1202.4427 [math.CO], 2012; Order 30.2 (2013): 427-435.

Sean A. Irvine, Java program (github)

J. Kahn, Entropy, independent sets and antichains: a new approach to Dedekind's problem, Proc. Amer. Math. Soc. 130 (2002), no. 2, 371-378.

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

Bjørn Kjos-Hanssen, Lei Liu, The number of languages with maximum state complexity, 2018.

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.

M. M. Krieger, Letter to N. J. A. Sloane, Jul 31 1975, confirming that a(7) = 2414682040998, using W. F. Lunnon's method but getting a different answer.

C. L. Mallows, Emails to N. J. A. Sloane, Jun-Jul 1991

Math. Stack Exchange, Counting antichains in the limit n->oo, 2014.

Muroga, Saburo, Iwao Toda, and Satoru Takasu, Theory of majority decision elements, Journal of the Franklin Institute 271.5 (1961): 376-418. [Annotated scans of pages 413 and 414 only]

R. A. Obando, Project: A map of a rule space.

Terry Speed, Letter to N. J. A. Sloane, Sep 20 1981.

Tamon Stephen and Timothy Yusun, Counting inequivalent monotone Boolean functions, arXiv preprint arXiv:1209.4623 [cs.DS], 2012.

V. G. Tkachenco, O. V. Sinyavsky, Blocks of Monotone Boolean Functions of Rank 5, Computer Science and Information Technology 4(4): 139-146, 2016; DOI: 10.13189/csit.2016.040402.

Tom Trotter, An Application of the Erdos/Stone Theorem, Sept. 13, 2001.

Eric Weisstein's World of Mathematics, Antichain

D. H. Wiedemann, Letter to N. J. A. Sloane, Nov 03, 1990

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

Gus Wiseman, Sequences enumerating clutters, antichains, hypertrees, and hyperforests, organized by labeling, spanning, and allowance of singletons.

R. Zeno, A007501 is an upper bound

V. D. Zolotarev, Enumeration of Boolean functions (Russian), Izvest. Vyssh. Uchebnykh Zavedenii Elektro. Novocherkassk, #3, 1970, 309-313; Math. Rev., 45#83, Jan. 1973.

Index entries for sequences related to Boolean functions


The asymptotics can be found in the Korshunov paper. - Boris Bukh, Nov 07 2003

a(n) = Sum_{k=1..n} binomial(n,k)*A006126(k) + 2, i.e., this sequence is the inverse binomial transform of A006126, plus 2. E.g. a(3) = 3*1 + 3*2 + 1*9 + 2 = 20. - Rodrigo A. Obando (R.Obando(AT)computer.org), Jul 26 2004


a(2)=6 from the antichains {}, {{}}, {{1}}, {{2}}, {{1,2}}, {{1},{2}}.

From Gus Wiseman, Feb 20 2019: (Start)

The a(0) = 2 through a(3) = 20 antichains:

  {}    {}     {}        {}

  {{}}  {{}}   {{}}      {{}}

        {{1}}  {{1}}     {{1}}

               {{2}}     {{2}}

               {{12}}    {{3}}

               {{1}{2}}  {{12}}


















stableSets[u_, Q_]:=If[Length[u]===0, {{}}, With[{w=First[u]}, Join[stableSets[DeleteCases[u, w], Q], Prepend[#, w]&/@stableSets[DeleteCases[u, r_/; r===w||Q[r, w]||Q[w, r]], Q]]]];

Table[Length[stableSets[Subsets[Range[n]], SubsetQ]], {n, 0, nn}] (* Gus Wiseman, Feb 20 2019 *)


Equals A014466 + 1, also A007153 + 2. Cf. A003182, A059119.

Cf. A006126, A006602, A261005, A293606, A293993, A304996, A305000, A305844, A306505, A317674, A319721, A320449, A321679.

Sequence in context: A168268 A277876 A002078 * A333445 A123930 A238895

Adjacent sequences:  A000369 A000370 A000371 * A000373 A000374 A000375




N. J. A. Sloane


a(8) from D. H. Wiedemann, personal communication, circa 1990

Additional comments from Michael Somos, Jun 10 2002



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Last modified September 24 17:09 EDT 2020. Contains 337321 sequences. (Running on oeis4.)