A304996 -id:A304996 - cp's OEIS frontend

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.

2, 3, 6, 20, 168, 7581, 7828354, 2414682040998, 56130437228687557907788, 286386577668298411128469151667598498812366

Offset: 0

A monotone Boolean function is an increasing function 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
From Amiram Eldar, May 28 2021 and Michel Marcus, Apr 07 2023: (Start)
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)

			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}}
                         {{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)

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.
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.

Frank a Campo, Relations between powers of Dedekind Numbers and exponential sums related to them, J. Int. Seq. Vol. 21 (2018), Article 18.4.4.
Frank a Campo, A Flexible Approach for the Enumeration of Down-Sets and its Application on Dedekind Numbers, arXiv:2206.10293 [math.CO], 2022.
Aureli Alabert, Mercè Farré, and Rubén Montes, Optimal Decision Rules for the Discursive Dilemma, arXiv:2210.13100 [math.OC], 2022.
J. M. Aranda, C program
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, Vol. 27, No. 1 (1979), pp. 1-9. MR0541338 (81b:05010)
Achille Basile, Anna De Simone, and Ciro Tarantino, A Note on Binary Strategy-Proof Social Choice Functions, Games (2022), Vol. 13, 78.
Ringo Baumann and Hannes Strass, On the number of bipolar Boolean functions, Journal of Logic and Computation, Vol. 27, No. 8 (2017), pp. 2431-2449; preprint.
Martin Berglund, Brink van der Merwe, and Steyn van Litsenborgh, Regular Expressions with Lookahead, J. Universal Comp. Sci. (2021) Vol. 27, No. 4, 324-340.
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.
Joel Berman and Peter Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, Vol. 121 (1976), pp. 103-124. [Annotated scanned copy]
J. Berman and P. Köhler, On Dedekind Numbers and Two Sequences of Knuth, J. Int. Seq., Vol. 24 (2021), Article 21.10.7.
Sara Billey and Matjaž Konvalinka, Generalized rank functions and quilts of alternating sign matrices, arXiv:2412.03236 [math.CO], 2024. See pp. 8-9.
Aniruddha Biswas and Palash Sarkar, Counting Unate and Monotone Boolean Functions Under Restrictions of Balancedness and Non-Degeneracy, J. Int. Seq. (2025) Vol. 28, Art. No. 25.3.4. See p. 2.
Stefan 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
Kevin S. Brown, Dedekind's problem.
Kevin S. Brown, Generating the Monotone Boolean Functions.
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
Claudine Chaouiya, Predo T. Monteiro, and Elisabeth Remy, Logical Modelling, Some Recent Methodological Advances Illustrated, Cellular Automata and Discrete Complex Systems: Proc. 30th IFIP WG 1.5 Int'l Wksp. (AUTOMATA 2024) Lect. Notes Comp. Sci. Vol 14782. Springer, Cham. See p. 8.
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]
Randolph Church, Numerical analysis of certain free distributive structures, Duke Math. J. 6 (1940). 732--734. [Scanned annotated copy]
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.
Jacob North Clark and Stephen Montgomery-Smith, Shapley-like values without symmetry, arXiv:1809.07747 [econ.TH], 2018.
Gábor Czédli, Minimum-sized generating sets of the direct powers of free distributive lattices, arXiv:2309.13783 [math.CO], 2023. See p. 16. See also CUBO, A Mathematical Journal Vol. 26, no. 2, pp. 217-237, August 2024.
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 and Stefan De Wannemacker, On the number of antichains of sets in a finite universe, arXiv:1407.4288 [math.CO], 2014 (see Table 1).
Patrick De Causmaecker, S. De Wannemacker, and J. Yellen, Intervals of Antichains and Their Decompositions, arXiv preprint arXiv:1602.04675 [math.CO], 2016.
Patrick De Causmaecker and Lennart Van Hirtum, Solving systems of equations on antichains for the computation of the ninth Dedekind Number, arXiv:2405.20904 [math.CO], 2024. See pp. 1, 3.
Richard Dedekind, Über Zerlegungen von Zahlen durch ihre grössten gemeinsamen Theiler, Festschrift Hoch. Braunschweig u. ges. Werke(II), 1897, pp. 103-148.; alternative link.
J. D. Farley, Was Gelfand right? The many loves of lattice theory, Notices AMS 69:2 (2022), 190-197.
Conor Finn and 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.
E. N. Gilbert, Lattice theoretic properties of frontal switching functions, J. Math. Phys., Vol. 33, No. 1-4, (1954), pp. 57-67, see Table III.
Milton W. Green, Letter to N. J. A. Sloane, 1973 (note "A360" refers to N0360 which is A000788).
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.
Dmitry I. Ignatov, A Note on Counting Basic Choice Functions with Formal Concept Analysis, 32nd Int'l Joint Conf. on Artif. Int., Formal Conc. Anal. Artif. Int. (IJCAI, FCA4AI 2023) 47-55.
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)
Christian Jäkel, A computation of the ninth Dedekind Number, Journal of Computational Algebra, Vol. 6-7, n° 100006 (doi:10.1016/j.jaca.2023.100006); also: arXiv:2304.00895 [math.CO], 2023.
J. Kahn, Entropy, independent sets and antichains: a new approach to Dedekind's problem, Proc. Amer. Math. Soc. 130 (2002), no. 2, 371-378.
Jonathan L. King, Brick tiling and monotone Boolean functions [Dead link, see next link].
Jonathan L. King, A change-of-coordinates from Geometry to Algebra, applied to Brick Tilings, arXiv:math/9809176 [math.CO], 1998.
Bjørn Kjos-Hanssen and 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.
Mathematics Stack Exchange, Counting antichains in the limit n->oo, 2014.
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]
J. B. Nation and Gianluca Paolini, Elementary properties of free lattices II: decidability of the universal theory, arXiv:2504.09128 [math.LO], 2025. See p. 25.
R. A. Obando, Project: A map of a rule space.
Bartlomiej Pawelski, On the number of inequivalent monotone Boolean functions of 8 variables, arXiv:2108.13997 [math.CO], 2021. See Table 2 p. 2.
Bartlomiej Pawelski, On the number of inequivalent monotone Boolean functions of 9 variables, arXiv:2305.06346 [math.CO], 2023.
Bartlomiej Pawelski and Andrzej Szepietowski, Divisibility properties of Dedekind numbers, arXiv:2302.04615 [math.CO], 2023.
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.
Andrzej Szepietowski, Fixes of permutations acting on monotone Boolean functions, arXiv:2205.03868 [math.CO], 2022. See p. 17.
V. G. Tkachenco and 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 Erdős/Stone Theorem, Sept. 13, 2001.
Lennart Van Hirtum, Patrick De Causmaecker, Jens Goemaere, Tobias Kenter, Heinrich Riebler, Michael Lass, and Christian Plessl, A computation of D(9) using FPGA Supercomputing, arXiv:2304.03039 [cs.DM], 2023.
Morgan Ward, Note on the order of free distributive lattices Bulletin of the American Mathematical Society, Vol. 52, No. 5 (1946), p. 423.
Eric Weisstein's World of Mathematics, Dedekind Number
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.
Wikipedia, Dedekind number.
Gus Wiseman, Sequences enumerating clutters, antichains, hypertrees, and hyperforests, organized by labeling, spanning, and allowance of singletons.
K. Yamamoto, Logarithmic order of free distributive lattice, Math. Soc. Japan, 6 (1954), 343-353.
R. Zeno, A007501 is an upper bound [Dead link]
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

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

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

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 *)

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
From J. M. Aranda, Jun 12 2021: (Start)
a(n) = A132581(2^n) = A132581(2^n-2^m) + A132581(2^n-2^(n-m)) for n >= m >= 0.
a(n) = A132582(3*2^n -1) for n >= 0.
(End)

a(8) from D. H. Wiedemann, personal communication, Nov 03 1990
Additional comments from Michael Somos, Jun 10 2002
a(9) from C. Jäkel added by Michel Marcus, Apr 04 2023

A014466 Dedekind numbers: monotone Boolean functions, or nonempty antichains of subsets of an n-set.

Original entry on oeis.org

1, 2, 5, 19, 167, 7580, 7828353, 2414682040997, 56130437228687557907787, 286386577668298411128469151667598498812365

Offset: 0

N. J. A. Sloane

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.
Also antichains of nonempty subsets of an n-set. The unlabeled case is A306505. The spanning case is A307249. This sequence has a similar description to A305000 except that the singletons must be disjoint from the other edges. - Gus Wiseman, Feb 20 2019
a(n) is the total number of hierarchical log-linear models on n labeled factors (categorical variables). See Wickramasinghe (2008) and Nardi and Rinaldo (2012). - Petros Hadjicostas, Apr 08 2020
From Lorenzo Sauras Altuzarra, Apr 02 2023: (Start)
a(n) is the number of labeled abstract simplicial complexes on n vertices.
A058673(n) <= a(n) <= A058891(n+1). (End)

			a(2)=5 from the antichains {{}}, {{1}}, {{2}}, {{1,2}}, {{1},{2}}.
From _Gus Wiseman_, Feb 20 2019: (Start)
The a(0) = 1 through a(3) = 19 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)
From _Lorenzo Sauras Altuzarra_, Apr 02 2023: (Start)
The 19 sets E such that ({1, 2, 3}, E) is an abstract simplicial complex:
  {}
  {{1}}
  {{2}}
  {{3}}
  {{1}, {2}}
  {{1}, {3}}
  {{2}, {3}}
  {{1}, {2}, {3}}
  {{1}, {2}, {1, 2}}
  {{1}, {3}, {1, 3}}
  {{2}, {3}, {2, 3}}
  {{1}, {2}, {3}, {1, 2}}
  {{1}, {2}, {3}, {1, 3}}
  {{1}, {2}, {3}, {2, 3}}
  {{1}, {2}, {3}, {1, 2}, {1, 3}}
  {{1}, {2}, {3}, {1, 2}, {2, 3}}
  {{1}, {2}, {3}, {1, 3}, {2, 3}}
  {{1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}}
  {{1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}
(End)

I. 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 27: 1-21 (1987).
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.
J. Dezert, Fondations pour une nouvelle théorie 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.
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.

Guillermo Alesandroni and Sungjin Kim, The probability that the Taylor resolution of a monomial ideal is minimal, J. Comm. Alg., 2024. See p. 3.
K. Atanassov, On Some of Smarandache's Problems.
K. S. Brown, Dedekind's problem.
Winfried Bruns, Pedro A. García-Sánchez, and Luca Moci, The monoid of monotone functions on a poset and arithmetic multiplicities for uniform matroids, arXiv:1902.00864 [math.CO], 2019.
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
David Carey and Mordechai Katzman, Stanley-Reisner Ideals with Pure Resolutions, arXiv:2409.05481 [math.AC], 2024. See p. 52.
Fan Cheng, Optimality of routing on the wiretap network with simple network topology, Information Theory (ISIT), 2014 IEEE International Symposium on, June 29 2014-July 4 2014 Page(s): 786 - 790 INSPEC Accession Number: 14524545 Honolulu, HI.
Fan Cheng and Vincent Y. F. Tan, A Numerical Study on the Wiretap Network with a Simple Network Topology, arXiv preprint arXiv:1505.02862 [cs.IT], 2015.
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.
J. L. King, A change-of-coordinates from Geometry to Algebra applied to brick tilings, arXiv:math/9809176 [math.CO], 1998.
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.
Y. Nardi and A. Rinaldo, The log-linear group-lasso estimator and its asymptotic properties, Bernoulli 18(3) (2012), 945-974; see Table 2 on p. 954.
Eric Weisstein's World of Mathematics, Antichain.
R. I. P. Wickramasinghe, Topics in log-linear models, Master of Science thesis in Statistics, Texas Tech University, Lubbock, TX, 2008. [For n = 2, the a(2) = 5 hierarchical log-linear models on two factors X and Y appear on p. 18. For n = 3, the a(3) = 19 hierarchical log-linear models on three factors X, Y, and Z, appear on p. 36. - _Petros Hadjicostas_, Apr 08 2020]
D. H. Wiedemann, A computation of the eighth Dedekind number, Order 8 (1991) 5-6.
Wikipedia, Abstract simplicial complex.
Gus Wiseman, Sequences enumerating clutters, antichains, hypertrees, and hyperforests, organized by labeling, spanning, and allowance of singletons.
Index entries for sequences related to Boolean functions

Equals A000372 - 1 = A007153 + 1.

Cf. A003182, A005465, A006126, A006602, A058673 (labeled matroids), A058891 (labeled hypergraphs), A261005, A293606, A304996, A305000, A306505, A307249, A317674, A319721, A320449, A321679.

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],{1,n}],SubsetQ]],{n,0,nn}] (* Gus Wiseman, Feb 20 2019 *)
A[s_Integer] := With[{s6 = StringPadLeft[ToString[s], 6, "0"]}, Cases[ Import["https://oeis.org/A" <> s6 <> "/b" <> s6 <> ".txt", "Table"], {, }][[All, 2]]];
A@372 - 1 (* Jean-François Alcover, Jan 07 2020 *)

Binomial transform of A307249 (or A006126 if its zeroth term is 1). - Gus Wiseman, Feb 20 2019

a(n) >= A005465(n) (because the hierarchical log-linear models on n factors always include all the conditional independence models considered by I. J. Good in A005465). - Petros Hadjicostas, Apr 24 2020

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

A003182 Dedekind numbers: inequivalent monotone Boolean functions of n or fewer variables, or antichains of subsets of an n-set.

Original entry on oeis.org

2, 3, 5, 10, 30, 210, 16353, 490013148, 1392195548889993358, 789204635842035040527740846300252680

Offset: 0

N. J. A. Sloane

NP-equivalence classes of unate Boolean functions of n or fewer variables.
Also the number of simple games with n players in minimal winning form up to isomorphism. - Fabián Riquelme, Mar 13 2018
The labeled case is A000372. - Gus Wiseman, Feb 23 2019
First differs from A348260(n + 1) at a(5) = 210, A348260(6) = 233. - Gus Wiseman, Nov 28 2021
Pawelski & Szepietowski show that a(n) = A001206(n) (mod 2) and that a(9) = 6 (mod 210). - Charles R Greathouse IV, Feb 16 2023

			From _Gus Wiseman_, Feb 20 2019: (Start)
Non-isomorphic representatives of the a(0) = 2 through a(3) = 10 antichains:
  {}    {}     {}         {}
  {{}}  {{}}   {{}}       {{}}
        {{1}}  {{1}}      {{1}}
               {{1,2}}    {{1,2}}
               {{1},{2}}  {{1},{2}}
                          {{1,2,3}}
                          {{1},{2,3}}
                          {{1},{2},{3}}
                          {{1,3},{2,3}}
                          {{1,2},{1,3},{2,3}}
(End)

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.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 273.
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.
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.
Saburo Muroga, Threshold Logic and Its Applications. Wiley, NY, 1971, p. 38, Table 2.3.2. - Row 13.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
D. H. Wiedemann, personal communication.

K. S. Brown, Dedekind's problem
Aniruddha Biswas and Palash Sarkar, Counting Unate and Monotone Boolean Functions Under Restrictions of Balancedness and Non-Degeneracy, J. Int. Seq. (2025) Vol. 28, Art. No. 25.3.4. See pp. 2, 14.
Patrick De Causmaecker and Stefan De Wannemacker, On the number of antichains of sets in a finite universe, arXiv:1407.4288 [math.CO], 2014.
Patrick De Causmaecker and Lennart Van Hirtum, Solving systems of equations on antichains for the computation of the ninth Dedekind Number, arXiv:2405.20904 [math.CO], 2024. See p. 4.
Liviu Ilinca and Jeff Kahn, Counting maximal antichains and independent sets, arXiv:1202.4427 [math.CO], 2012; Order 30.2 (2013): 427-435.
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.
Sascha Kurz, Competitive learning of monotone Boolean functions, arXiv:1401.8135 [cs.DS], 2014.
C. L. Mallows, Emails to N. J. A. Sloane, Jun-Jul 1991
Mikaël Monet and Dan Olteanu, Towards Deterministic Decomposable Circuits for Safe Queries, 2018.
S. Muroga, Threshold Logic and Its Applications, Wiley, NY, 1971 [Annotated scans of a few pages]
Bartlomiej Pawelski, On the number of inequivalent monotone Boolean functions of 8 variables, arXiv:2108.13997 [math.CO], 2021. See Table 2 p. 2.
Bartlomiej Pawelski, On the number of inequivalent monotone Boolean functions of 9 variables, arXiv:2305.06346 [math.CO], 2023.
Bartłomiej Pawelski, Counting and generating monotone Boolean functions, Doctoral Diss., Univ. Gdańsk, (Poland, 2024). See pp. 12, 24, 34-35, 40, 49-50.
Bartlomiej Pawelski and Andrzej Szepietowski, Divisibility properties of Dedekind numbers, arXiv:2302.04615 [math.CO], 2023.
Tamon Stephen and Timothy Yusun, Counting inequivalent monotone Boolean functions, Discrete Applied Mathematics, 167 (2014), 15-24.
Tamon Stephen and Timothy Yusun, Counting inequivalent monotone Boolean functions, arXiv preprint arXiv:1209.4623 [cs.DS], 2012.
Andrzej Szepietowski, Fixes of permutations acting on monotone Boolean functions, arXiv:2205.03868 [math.CO], 2022. See p. 17.
Eric Weisstein's World of Mathematics, Boolean Function.
Gus Wiseman, Sequences enumerating clutters, antichains, hypertrees, and hyperforests, organized by labeling, spanning, and allowance of singletons.
Index entries for sequences related to Boolean functions

Cf. A000372, A007153, A006602, A007411.
Cf. A006126, A014466, A261005, A293606, A293993, A304996, A305000, A305857, A306505, A319721, A320449, A321679.
Cf. A007363, A046165, A306007, A307249, A326358, A326363.

a(n) = A306505(n) + 1. - Gus Wiseman, Jul 02 2019

a(7) added by Timothy Yusun, Sep 27 2012
a(8) from Pawelski added by Michel Marcus, Sep 01 2021
a(9) from Pawelski added by Michel Marcus, May 11 2023

A305000 Number of labeled antichains of finite sets spanning some subset of {1,...,n} with singleton edges allowed.

Original entry on oeis.org

1, 2, 8, 72, 1824, 220608, 498243968, 309072306743552, 14369391925598802012151296, 146629927766168786239127150948525247729660416

Offset: 0

Gus Wiseman, May 23 2018

Only the non-singleton edges are required to form an antichain.

Number of non-degenerate unate Boolean functions of n or fewer variables. - Aniruddha Biswas, May 11 2024

			The a(2) = 8 antichains:
  {}
  {{1}}
  {{2}}
  {{1,2}}
  {{1},{2}}
  {{1},{1,2}}
  {{2},{1,2}}
  {{1},{2},{1,2}}

Aniruddha Biswas and Palash Sarkar, Counting unate and balanced monotone Boolean functions, arXiv:2304.14069 [math.CO], 2023.
Aniruddha Biswas and Palash Sarkar, Counting Unate and Monotone Boolean Functions Under Restrictions of Balancedness and Non-Degeneracy, J. Int. Seq. (2025) Vol. 28, Art. No. 25.3.4. See pp. 4, 12.
Gus Wiseman, Sequences enumerating clutters, antichains, hypertrees, and hyperforests, organized by labeling, spanning, and allowance of singletons.

Cf. A006126, A261005, A304996, A304997, A304998, A304999, A305000, A305001.
Cf. A000372, A003182, A014466, A293606, A293993, A306505, A320449, A321679.
Cf. A245079.

Binomial transform of A304999.

Inverse binomial transform of A245079. - Aniruddha Biswas, May 11 2024

a(5)-a(8) from Gus Wiseman, May 31 2018

a(9) from Aniruddha Biswas, May 11 2024

A305001 Number of labeled antichains of finite sets spanning n vertices without singletons.

Original entry on oeis.org

1, 0, 1, 5, 87, 6398, 7745253, 2414573042063, 56130437190053518791691, 286386577668298410118121281898931424413687

Offset: 0

Gus Wiseman, May 23 2018

nonn

From Gus Wiseman, Jul 03 2019: (Start)
Also the number of antichains covering n vertices and having empty intersection (meaning there is no vertex in common to all the edges). For example, the a(3) = 5 antichains are:
  {{3},{1,2}}
  {{2},{1,3}}
  {{1},{2,3}}
  {{1},{2},{3}}
  {{1,2},{1,3},{2,3}}
(End)

			The a(3) = 5 antichains:
  {{1,2,3}}
  {{1,2},{1,3}}
  {{1,2},{2,3}}
  {{1,3},{2,3}}
  {{1,2},{1,3},{2,3}}

The binomial transform is the non-covering case A307249.
The second binomial transform is A014466.
Cf. A000372, A003182, A006126, A006602, A046165, A261005, A304996, A304997, A304998, A304999, A305000, A326358, A326359.

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[Select[stableSets[Subsets[Range[n],{1,n}],SubsetQ],And[Union@@#==Range[n],#=={}||Intersection@@#=={}]&]],{n,0,5}] (* Gus Wiseman, Jul 03 2019 *)

a(9) from A307249 - Dmitry I. Ignatov, Nov 27 2023

A304998 Number of unlabeled antichains of finite sets spanning n vertices without singletons.

Original entry on oeis.org

1, 0, 1, 3, 15, 160, 15963, 489980652

Offset: 0

Gus Wiseman, May 23 2018

nonn

			Non-isomorphic representatives of the a(4) = 15 antichains:
  {{1,2,3,4}}
  {{1,2},{3,4}}
  {{1,4},{2,3,4}}
  {{1,3,4},{2,3,4}}
  {{1,2},{1,3,4},{2,3,4}}
  {{1,3},{1,4},{2,3,4}}
  {{1,3},{2,4},{3,4}}
  {{1,4},{2,4},{3,4}}
  {{1,2,4},{1,3,4},{2,3,4}}
  {{1,2},{1,3},{1,4},{2,3,4}}
  {{1,2},{1,3},{2,4},{3,4}}
  {{1,4},{2,3},{2,4},{3,4}}
  {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}
  {{1,3},{1,4},{2,3},{2,4},{3,4}}
  {{1,2},{1,3},{1,4},{2,3},{2,4},{3,4}}

Cf. A006126, A261005, A304996, A304997, A304998, A304999, A305000, A305001.

a(n > 0) = A261005(n) - A261005(n - 1).

A304997 Number of unlabeled antichains of finite sets spanning n vertices with singleton edges allowed.

Original entry on oeis.org

1, 1, 4, 18, 142, 3100, 823042

Offset: 0

Gus Wiseman, May 23 2018

			Non-isomorphic representatives of the a(3) = 18 antichains:
{{1,2,3}}
{{3},{1,2}}
{{3},{1,2,3}}
{{1,3},{2,3}}
{{1},{2},{3}}
{{2},{3},{1,3}}
{{2},{3},{1,2,3}}
{{3},{1,2},{2,3}}
{{3},{1,3},{2,3}}
{{1,2},{1,3},{2,3}}
{{1},{2},{3},{2,3}}
{{1},{2},{3},{1,2,3}}
{{2},{3},{1,2},{1,3}}
{{2},{3},{1,3},{2,3}}
{{3},{1,2},{1,3},{2,3}}
{{1},{2},{3},{1,3},{2,3}}
{{2},{3},{1,2},{1,3},{2,3}}
{{1},{2},{3},{1,2},{1,3},{2,3}}

Cf. A006126, A261005, A304996, A304997, A304998, A304999, A305000, A305001.

A306505 Number of non-isomorphic antichains of nonempty subsets of {1,...,n}.

Original entry on oeis.org

1, 2, 4, 9, 29, 209, 16352, 490013147, 1392195548889993357, 789204635842035040527740846300252679

Offset: 0

Gus Wiseman, Feb 20 2019

The spanning case is A006602 or A261005. The labeled case is A014466.
From Petros Hadjicostas, Apr 22 2020: (Start)
a(n) is the number of "types" of log-linear hierarchical models on n factors in the sense of Colin Mallows (see the emails to N. J. A. Sloane).
Two hierarchical models on n factors belong to the same "type" iff one can obtained from the other by a permutation of the factors.
The total number of hierarchical log-linear models on n factors (in all "types") is given by A014466(n) = A000372(n) - 1.
The name of a hierarchical log-linear model on factors is based on the collection of maximal interaction terms, which must be an antichain (by the definition of maximality).
In his example on p. 1, Colin Mallows groups the A014466(3) = 19 hierarchical log-linear models on n = 3 factors x, y, z into a(3) = 9 types. See my example below for more details. (End)
First differs from A348260(n + 1) - 1 at a(5) = 209, A348260(6) - 1 = 232. - Gus Wiseman, Nov 28 2021

			Non-isomorphic representatives of the a(0) = 1 through a(3) = 9 antichains:
  {}  {}     {}         {}
      {{1}}  {{1}}      {{1}}
             {{1,2}}    {{1,2}}
             {{1},{2}}  {{1},{2}}
                        {{1,2,3}}
                        {{1},{2,3}}
                        {{1},{2},{3}}
                        {{1,3},{2,3}}
                        {{1,2},{1,3},{2,3}}
From _Petros Hadjicostas_, Apr 23 2020: (Start)
We expand _Colin Mallows_'s example from p. 1 of his list of 1991 emails. For n = 3, we have the following a(3) = 9 "types" of log-linear hierarchical models:
Type 1: ( ), Type 2: (x), (y), (z), Type 3: (x,y), (y,z), (z,x), Type 4: (x,y,z), Type 5: (xy), (yz), (zx), Type 6: (xy,z), (yz,x), (zx,y), Type 7: (xy,xz), (yx,yz), (zx,zy), Type 8: (xy,yz,zx), Type 9: (xyz).
For each model, the name only contains the maximal terms. See p. 36 in Wickramasinghe (2008) for the full description of the 19 models.
Strictly speaking, I should have used set notation (rather than parentheses) for the name of each model, but I follow the tradition of the theory of log-linear models. In addition, in an interaction term such as xy, the order of the factors is irrelevant.
Models in the same type essentially have similar statistical properties.
For example, models in Type 7 have the property that two factors are conditionally independent of one another given each level (= category) of the third factor.
Models in Type 6 are such that two factors are jointly independent from the third one. (End)

C. L. Mallows, Emails to N. J. A. Sloane, Jun-Jul 1991, p. 1.
R. I. P. Wickramasinghe, Topics in log-linear models, Master of Science thesis in Statistics, Texas Tech University, Lubbock, TX, 2008, p. 36.
Gus Wiseman, Sequences enumerating clutters, antichains, hypertrees, and hyperforests, organized by labeling, spanning, and allowance of singletons.

Cf. A000372, A003182, A006126, A006602, A014466, A261005, A293606, A293993, A304996, A305000, A305001, A305857, A317674, A319721, A320449, A321679.

Cf. A007363, A306007, A307249, A326358, A326359, A326360, A326363.

a(n) = A003182(n) - 1.

Partial sums of A006602 minus 1.

a(8) from A003182. - Bartlomiej Pawelski, Nov 27 2022

a(9) from A003182. - Dmitry I. Ignatov, Nov 27 2023

cp's OEIS Frontend

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.

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A014466 Dedekind numbers: monotone Boolean functions, or nonempty antichains of subsets of an n-set.

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A003182 Dedekind numbers: inequivalent monotone Boolean functions of n or fewer variables, or antichains of subsets of an n-set.

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A305000 Number of labeled antichains of finite sets spanning some subset of {1,...,n} with singleton edges allowed.

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A305001 Number of labeled antichains of finite sets spanning n vertices without singletons.

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A304998 Number of unlabeled antichains of finite sets spanning n vertices without singletons.

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A304997 Number of unlabeled antichains of finite sets spanning n vertices with singleton edges allowed.

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A306505 Number of non-isomorphic antichains of nonempty subsets of {1,...,n}.

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