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A000372
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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)
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100
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OFFSET
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0,1
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COMMENTS
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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 terms were first calculated by:
a(0)-a(4) - Dedekind (1897)
a(5) - Church (1940)
a(6) - Ward (1946)
a(7) - Church (1965, verified by Berman and Kohler, 1976)
a(8) - Wiedemann (1991)
a(9) - Jäkel (2023)
a(9) - independently computed by Lennart Van Hirtum, Patrick De Causmaecker, Jens Goemaere, Tobias Kenter, Heinrich Riebler, Michael Lass, and Christian Plessl (2023)
(End)
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REFERENCES
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Ian Anderson, Combinatorics of Finite Sets. Oxford Univ. Press, 1987, p. 38.
Jorge Luis Arocha, Antichains in ordered sets [in Spanish], Anales del Instituto de Matematicas de la Universidad Nacional Autonoma de Mexico, Vol. 27 (1987), pp. 1-21.
Joel Berman and Peter Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, Vol. 121 (1976), pp. 103-124.
Garrett Birkhoff, Lattice Theory, American Mathematical Society, Colloquium Publications, Vol. 25, 3rd ed., Providence, RI, 1967, p. 63.
Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 273.
J. D. Farley, Was Gelfand right? The many loves of lattice theory, Notices AMS 69:2 (2022),190-197.
Michael A. Harrison, Introduction to Switching and Automata Theory, McGraw Hill, NY, 1965, p. 188.
Donald 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, in D. J. A. Welsh, editor, Combinatorial Mathematics and Its Applications. Academic Press, NY, 1971, pp. 173-181.
Saburo Muroga, Threshold Logic and Its Applications. Wiley, NY, 1971, pp. 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).
Douglas B. West, Introduction to Graph Theory, 2nd ed., Prentice-Hall, NJ, 2001, p. 349.
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LINKS
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Joel 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, Springer, Berlin, Heidelberg, 1983, pp. 10-53.
Randolph Church, Enumeration by rank of the free distributive lattice with seven generators, Notices of the American Mathematical Society, Vol. 12, No. 6 (1965), p. 724; entire volume.
Saburo Muroga, 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]
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FORMULA
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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(n) = A132582(3*2^n -1) for n >= 0.
(End)
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EXAMPLE
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a(2)=6 from the antichains {}, {{}}, {{1}}, {{2}}, {{1,2}}, {{1},{2}}.
The a(0) = 2 through a(3) = 20 antichains:
{} {} {} {}
{{}} {{}} {{}} {{}}
{{1}} {{1}} {{1}}
{{2}} {{2}}
{{12}} {{3}}
{{1}{2}} {{12}}
{{13}}
{{23}}
{{123}}
{{1}{2}}
{{1}{3}}
{{2}{3}}
{{1}{23}}
{{2}{13}}
{{3}{12}}
{{12}{13}}
{{12}{23}}
{{13}{23}}
{{1}{2}{3}}
{{12}{13}{23}}
(End)
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MATHEMATICA
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nn=5;
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 *)
Table[Total[Boole[Table[UnateQ[BooleanFunction[k, n]], {k, 0, 2^(2^n) - 1}]]], {n, 0, 4}] (* Eric W. Weisstein, Jun 27 2023 *)
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CROSSREFS
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Cf. A006126, A006602, A261005, A293606, A293993, A304996, A305000, A305844, A306505, A317674, A319721, A320449, A321679.
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KEYWORD
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nonn,hard,more,nice
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AUTHOR
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EXTENSIONS
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a(8) from D. H. Wiedemann, personal communication, Nov 03 1990
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STATUS
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approved
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