A006126 -id:A006126 - 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

A285572 Number of finite sets of pairwise indivisible positive integers with least common multiple n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 1, 3, 1, 9, 1, 1, 2, 2, 2, 6, 1, 2, 2, 4, 1, 9, 1, 3, 3, 2, 1, 5, 1, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 23, 1, 2, 3, 1, 2, 9, 1, 3, 2, 9, 1, 10, 1, 2, 3, 3, 2, 9, 1, 5, 1, 2, 1, 23, 2, 2, 2, 4, 1, 23, 2, 3, 2, 2, 2, 6, 1, 3, 3, 6

Offset: 1

Gus Wiseman, Apr 21 2017

nonn

			The a(72)=10 sets are {72}, {8,9}, {8,18}, {8,36}, {9,24}, {18,24}, {24,36}, {6,8,9}, {8,9,12}, {8,12,18}.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000666, A006126, A048143, A054921, A076078, A076413, A285573.

nn=50;
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[Rest[stableSets[Divisors[n],Divisible]],LCM@@#===n&]],{n,1,nn}]

A367903 Number of sets of nonempty subsets of {1..n} contradicting a strict version of the axiom of choice.

Original entry on oeis.org

0, 0, 1, 67, 30997, 2147296425, 9223372036784737528, 170141183460469231731687303625772608225, 57896044618658097711785492504343953926634992332820282019728791606173188627779

Offset: 0

Gus Wiseman, Dec 05 2023

The axiom of choice says that, given any set of nonempty sets Y, it is possible to choose a set containing an element from each. The strict version requires this set to have the same cardinality as Y, meaning no element is chosen more than once.

			The a(2) = 1 set-system is {{1},{2},{1,2}}.
The a(3) = 67 set-systems have following 21 non-isomorphic representatives:
  {{1},{2},{1,2}}
  {{1},{2},{3},{1,2}}
  {{1},{2},{3},{1,2,3}}
  {{1},{2},{1,2},{1,3}}
  {{1},{2},{1,2},{1,2,3}}
  {{1},{2},{1,3},{2,3}}
  {{1},{2},{1,3},{1,2,3}}
  {{1},{1,2},{1,3},{2,3}}
  {{1},{1,2},{1,3},{1,2,3}}
  {{1},{1,2},{2,3},{1,2,3}}
  {{1,2},{1,3},{2,3},{1,2,3}}
  {{1},{2},{3},{1,2},{1,3}}
  {{1},{2},{3},{1,2},{1,2,3}}
  {{1},{2},{1,2},{1,3},{2,3}}
  {{1},{2},{1,2},{1,3},{1,2,3}}
  {{1},{2},{1,3},{2,3},{1,2,3}}
  {{1},{1,2},{1,3},{2,3},{1,2,3}}
  {{1},{2},{3},{1,2},{1,3},{2,3}}
  {{1},{2},{3},{1,2},{1,3},{1,2,3}}
  {{1},{2},{1,2},{1,3},{2,3},{1,2,3}}
  {{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}

Wikipedia, Axiom of choice.

Multisets of multisets of this type are ranked by A355529.
The version without singletons is A367769.
The version for simple graphs is A367867, covering A367868.
The version allowing empty edges is A367901.
The complement is A367902, without singletons A367770, ranks A367906.
For a unique choice (instead of none) we have A367904, ranks A367908.
These set-systems have ranks A367907.
An unlabeled version is A368094, for multiset partitions A368097.
A000372 counts antichains, covering A006126, nonempty A014466.
A003465 counts covering set-systems, unlabeled A055621.
A058891 counts set-systems, unlabeled A000612.
A059201 counts covering T_0 set-systems.
A323818 counts covering connected set-systems.
A326031 gives weight of the set-system with BII-number n.
Cf. A007716, A083323, A092918, A102896, A283877, A306445, A355739, A355740, A367862, A367905, A368409, A368413.

Table[Length[Select[Subsets[Rest[Subsets[Range[n]]]], Select[Tuples[#],UnsameQ@@#&]=={}&]],{n,0,3}]

a(n) + A367904(n) + A367772(n) = A058891(n+1) = 2^(2^n-1).

a(5)-a(8) from Christian Sievers, Jul 26 2024

A367902 Number of sets of nonempty subsets of {1..n} satisfying a strict version of the axiom of choice.

Original entry on oeis.org

1, 2, 7, 61, 1771, 187223, 70038280, 90111497503, 397783376192189

Offset: 0

Gus Wiseman, Dec 05 2023

			The a(2) = 7 set-systems:
  {}
  {{1}}
  {{2}}
  {{1,2}}
  {{1},{2}}
  {{1},{1,2}}
  {{2},{1,2}}

Wikipedia, Axiom of choice.

The version for simple graphs is A133686, covering A367869.
The version without singletons is A367770.
The complement allowing empty edges is A367901.
The complement is A367903, without singletons A367769, ranks A367907.
For a unique choice we have A367904, ranks A367908.
These set-systems have ranks A367906.
A000372 counts antichains, covering A006126, nonempty A014466.
A003465 counts covering set-systems, unlabeled A055621.
A058891 counts set-systems, unlabeled A000612.
A059201 counts covering T_0 set-systems.
A323818 counts covering connected set-systems.
A326031 gives weight of the set-system with BII-number n.
Cf. A007716, A083323, A092918, A102896, A283877, A306445, A355739, A355740, A367862, A367905, A370636.

Table[Length[Select[Subsets[Subsets[Range[n]]], Select[Tuples[#],UnsameQ@@#&]!={}&]],{n,0,3}]

a(n) = A370636(2^n-1). - Alois P. Heinz, Mar 09 2024

a(6)-a(8) from Christian Sievers, Jul 25 2024

A303362 Number of strict integer partitions of n with pairwise indivisible parts.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 2, 3, 4, 5, 4, 6, 7, 7, 9, 11, 12, 13, 15, 17, 20, 23, 25, 27, 32, 35, 40, 45, 50, 55, 58, 67, 78, 84, 95, 101, 113, 124, 137, 153, 169, 180, 198, 219, 242, 268, 291, 319, 342, 374, 412, 450, 492, 535, 573, 632, 685, 746, 813, 868, 944

Offset: 1

Gus Wiseman, Apr 22 2018

nonn

			The a(14) = 7 strict integer partitions are (14), (11,3), (10,4), (9,5), (8,6), (7,5,2), (7,4,3).

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..450, (terms up to a(250) from Andrew Howroyd)

Cf. A000009, A000837, A003238, A006126, A051424, A259936, A275307, A281116, A285572, A285573, A290103, A293606, A293993, A303364.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Select[Tuples[#,2],UnsameQ@@#&&Divisible@@#&]==={}&]],{n,60}]

lista(nn)={local(Cache=Map());
  my(excl=vector(nn, n, sumdiv(n, d, 2^(n-d))));
  my(a(n, m=n, b=0)=
     if(n==0, 1,
        while(m>n || bittest(b,0), m--; b>>=1);
        my(hk=[n, m, b], z);
        if(!mapisdefined(Cache, hk, &z),
          z = if(m, self()(n, m-1, b>>1) + self()(n-m, m, bitor(b, excl[m])), 0);
          mapput(Cache, hk, z)); z));
   for(n=1, nn, print1(a(n), ", "))
} \\ Andrew Howroyd, Nov 02 2019

A016031 De Bruijn's sequence: 2^(2^(n-1) - n): number of ways of arranging 2^n bits in circle so all 2^n consecutive strings of length n are distinct.

Original entry on oeis.org

1, 1, 2, 16, 2048, 67108864, 144115188075855872, 1329227995784915872903807060280344576, 226156424291633194186662080095093570025917938800079226639565593765455331328

Offset: 1

N. J. A. Sloane, Robert G. Wilson v

Sequence corresponds also to the largest number that may be determined by asking no more than 2^(n-1) - 1 questions("Yes"-or-"No" answerable) with lying allowed at most once. - Lekraj Beedassy, Jul 15 2002
Number of Ouroborean rings for binary n-tuplets. - Lekraj Beedassy, May 06 2004
Also the number of games of Nim that are wins for the second player when the starting position is either the empty heap or heaps of sizes 1 <= t_1 < t_2 < ... < t_k < 2^(n-1) (if n is 1, the only starting position is the empty heap). E.g.: a(4) = 16: the winning positions for the second player when all the heap sizes are different and less than 2^3: (4,5,6,7), (3,5,6), (3,4,7), (2,5,7), (2,4,6), (2,3,6,7), (2,3,4,5), (1,6,7), (1,4,5), (1,3,5,7), (1,3,4,6), (1,2,5,6), (1,2,4,7), (1,2,3), (1,2,3,4,5,6,7) and the empty heap. - Kennan Shelton (kennan.shelton(AT)gmail.com), Apr 14 2006
a(n + 1) = 2^(2^n-n-1) = 2^A000295(n) is also the number of set-systems on n vertices with no singletons. The case with singletons is A058891. The unlabeled case is A317794. The spanning/covering case is A323816. The antichain case is A006126. The connected case is A323817. The uniform case is A306021(n) - 1. The graphical case is A006125. The chain case is A005840. - Gus Wiseman, Feb 01 2019
Named after the Dutch mathematician Nicolaas Govert de Bruijn (1918-2012). - Amiram Eldar, Jun 20 2021

Jonathan L. Gross and Jay Yellen, eds., Handbook of Graph Theory, CRC Press, 2004, p. 255.
Frank Harary and Edgar M. Palmer, Graphical Enumeration, 1973, p. 31.
D. J. Newman, "A variation of the Game of Twenty Question", in: Murray S. Klamkin (ed.), Problems in Applied Mathematics, Philadelphia, PA: SIAM, 1990, Problem 66-20, pp. 121-122.
Richard P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Cor. 5.6.15.
Ian Stewart, Game, Set and Math, pp. 50, Penguin 1991.

Vincenzo Librandi, Table of n, a(n) for n = 1..12
CombOS - Combinatorial Object Server, Generate de Bruijn sequences.
Robert Erra, Nik Lygeros and Nigel Stewart, On Minimal Strings Containing the Elements of S(n) by Decimation, Proceedings AA (DM-CCG), 2001, Section 5.4.
Loïc Foissy, Hopf algebraic structures on hypergraphs and multi-complexes, arXiv:2304.00810 [math.CO], 2023.
Francisco J. Muñoz and Juan Carlos Nuño, Rule-based Generation of de Bruijn Sequences: Memory and Learning, arXiv:2507.09764 [cs.FL], 2025. See p. 9.
Wikipedia, De Bruijn sequence.

Cf. A000295, A003465, A006125, A058891 (set systems), A317794 (unlabeled case), A323816 (spanning case), A323817 (connected case), A331691 (alternating signs).

[2^(2^(n-1)-n): n in [1..10]]; // Vincenzo Librandi, Aug 09 2017

Table[2^(2^(n - 1) - n), {n, 20}] (* Vincenzo Librandi, Aug 09 2017 *)

a(n) = 2^A000295(n-1). - Lekraj Beedassy, Jan 17 2007

Shifting once to the left gives the binomial transform of A323816. - Gus Wiseman, Feb 01 2019

cp's OEIS Frontend

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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.

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

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A285572 Number of finite sets of pairwise indivisible positive integers with least common multiple n.

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A367903 Number of sets of nonempty subsets of {1..n} contradicting a strict version of the axiom of choice.

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A367902 Number of sets of nonempty subsets of {1..n} satisfying a strict version of the axiom of choice.

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A303362 Number of strict integer partitions of n with pairwise indivisible parts.

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PARI

A016031 De Bruijn's sequence: 2^(2^(n-1) - n): number of ways of arranging 2^n bits in circle so all 2^n consecutive strings of length n are distinct.