A305844 -id:A305844 - cp's OEIS frontend

A006126 Number of hierarchical models on n labeled factors or variables with linear terms forced. Also number of antichain covers of a labeled n-set.

2, 1, 2, 9, 114, 6894, 7785062, 2414627396434, 56130437209370320359966, 286386577668298410623295216696338374471993

Offset: 0

An antichain cover is a cover such that no element of the cover is a subset of another element of the cover.
Also, the number of nondegenerate monotone Boolean functions of n variables in an n-variable Boolean algebra. - Rodrigo A. Obando (R.Obando(AT)computer.org), Jul 26 2004
Also, number of simplicial complexes on an n-element vertex set. - Richard Stanley, Feb 10 2019
There are two antichains of size zero, namely {} and {{}}, while there is only one simplicial complex, namely {}. The unlabeled case is A006602. The non-covering case is A000372, which is A014466 plus 1. - Gus Wiseman, Mar 31 2019
From Petros Hadjicostas, Apr 10 2020: (Start)
Hierarchical models are always nonempty because they always include an intercept (or overall effect).
The total number of log-linear hierarchical models on n labeled factors (categorical variables) with no forcing of terms is given by A000372(n) - 1 (Dedekind numbers minus 1).
Hierarchical log-linear models for analyzing contingency tables are defined in the classic book by Bishop, Fienberg, and Holland (1975). (End)

			a(5) = 1 + 90 + 790 + 1895 + 2116 + 1375 + 490 + 115 + 20 + 2 = 6894.
There are 9 antichain covers of a labeled 3-set: {{1,2,3}}, {{1},{2,3}}, {{2},{1,3}}, {{3},{1,2}}, {{1,2},{1,3}}, {{1,2},{2,3}}, {{1,3},{2,3}}, {{1},{2},{3}}, {{1,2},{1,3},{2,3}}.
From _Gus Wiseman_, Feb 23 2019: (Start)
The a(0) = 2 through a(3) = 9 antichains:
  {}    {{1}}  {{12}}    {{123}}
  {{}}         {{1}{2}}  {{1}{23}}
                         {{2}{13}}
                         {{3}{12}}
                         {{12}{13}}
                         {{12}{23}}
                         {{13}{23}}
                         {{1}{2}{3}}
                         {{12}{13}{23}}
(End)

Y. M. M. Bishop, S. E. Fienberg and P. W. Holland, Discrete Multivariate Analysis. MIT Press, 1975, p. 34. [In part (e), the Hierarchy Principle for log-linear models is defined. It essentially says that if a higher-order parameter term is included in the log-linear model, then all the lower-order parameter terms should also be included. - Petros Hadjicostas, Apr 08 2020]
V. Jovovic and G. Kilibarda, On enumeration of the class of all monotone Boolean functions, in preparation.
C. L. Mallows, personal communication.
A. A. Mcintosh, personal communication.
R. A. Obando, On the number of nondegenerate monotone boolean functions of n variables, In Preparation.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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.
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.
Florian Bridoux, Amélia Durbec, Kévin Perrot, and Adrien Richard, Complexity of fixed point counting problems in Boolean Networks, arXiv:2012.02513 [math.CO], 2020.
Florian Bridoux, Nicolas Durbec, Kevin Perrot, and Adrien Richard, Complexity of Maximum Fixed Point Problem in Boolean Networks, Conference on Computability in Europe (CiE 2019) Computing with Foresight and Industry (Lecture Notes in Computer Science book series, Vol. 11558), Springer, Cham, 132-143.
K. S. Brown, Dedekind's problem
Patrick De Causmaecker and Stefan De Wannemacker, On the number of antichains of sets in a finite universe, arXiv:1407.4288 [math.CO], 2014.
V. Jovovic and G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, Diskretnaya Matematika, 11 (1999), no. 4, 127-138 (translated in Discrete Mathematics and Applications, 9, (1999), no. 6).
C. L. Mallows, Emails to N. J. A. Sloane, Jun-Jul 1991
C. L. Mallows & N. J. A. Sloane, Emails, May 1991
C. L. Mallows & N. J. A. Sloane, Emails, Jun. 1991
Eric Weisstein's World of Mathematics, Antichain.
Eric Weisstein's World of Mathematics, Cover.
R. I. P. Wickramasinghe, Topics in log-linear models, Master of Science thesis in Statistics, Texas Tech University, Lubbock, TX, 2008. [From the A000372(2) - 1 = 4 hierarchical log-linear models on two factors X and Y, on p. 18 of his thesis, only Models 11 and 15 force all the linear terms (i.e., a(2) = 2). From the A000372(3) - 1 = 19 hierarchical log-linear models on three factors X, Y, and Z, on p. 36 of his thesis, only Models 11-19 force all the linear terms (i.e., a(3) = 9). - _Petros Hadjicostas_, Apr 08 2020]
D. H. Wiedemann, Letter to N. J. A. Sloane, Nov 03, 1990
D. H. Wiedermann, Email to N. J. A. Sloane, May 28 1991
Gus Wiseman, Sequences enumerating clutters, antichains, hypertrees, and hyperforests, organized by labeling, spanning, and allowance of singletons.

Cf. A000372, A056046-A056049, A056052, A056101, A056104, A051112-A051118.

Cf. A006602, A014466, A261005, A293606, A293993, A305000, A305844, A306550, A307249, A317674, A319721, A320449.

nn=4;
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]],SubsetQ],Union@@#==Range[n]&]],{n,0,nn}] (* Gus Wiseman, Feb 23 2019 *)
A000372 = Cases[Import["https://oeis.org/A000372/b000372.txt", "Table"], {, }][[All, 2]];
lg = Length[A000372];
a372[n_] := If[0 <= n <= lg-1, A000372[[n+1]], 0];
a[n_] := Sum[(-1)^(n-k+1) Binomial[n, k-1] a372[k-1], {k, 0, lg}];
a /@ Range[0, lg-1] (* Jean-François Alcover, Jan 07 2020 *)

a(n) = Sum_{k = 1..C(n, floor(n/2))} b(k, n), where b(k, n) is the number of k-antichain covers of a labeled n-set.

Inverse binomial transform of A000372. - Gus Wiseman, Feb 24 2019

Last 3 terms from Michael Bulmer (mrb(AT)maths.uq.edu.au)
Antichain interpretation from Vladeta Jovovic and Goran Kilibarda, Jul 31 2000
a(0) = 2 added by Gus Wiseman, Feb 23 2019
Name edited by Petros Hadjicostas, Apr 08 2020
a(9) using A000372 added by Bruno L. O. Andreotti, May 14 2023

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.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

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

A305843 Number of labeled spanning intersecting set-systems on n vertices.

Original entry on oeis.org

1, 1, 3, 27, 1245, 1308285, 912811093455, 291201248260060977862887, 14704022144627161780742038728709819246535634969, 12553242487940503914363982718112298267975272588471811456164576678961759219689708372356843289

Offset: 0

Gus Wiseman, Jun 11 2018

nonn

An intersecting set-system S is a finite set of finite nonempty sets (edges), any two of which have a nonempty intersection. S is spanning if every vertex is contained in some edge.

			The a(3) = 27 spanning intersecting set-systems:
{{1,2,3}}
{{1},{1,2,3}}
{{2},{1,2,3}}
{{3},{1,2,3}}
{{1,2},{1,3}}
{{1,2},{2,3}}
{{1,2},{1,2,3}}
{{1,3},{2,3}}
{{1,3},{1,2,3}}
{{2,3},{1,2,3}}
{{1},{1,2},{1,3}}
{{1},{1,2},{1,2,3}}
{{1},{1,3},{1,2,3}}
{{2},{1,2},{2,3}}
{{2},{1,2},{1,2,3}}
{{2},{2,3},{1,2,3}}
{{3},{1,3},{2,3}}
{{3},{1,3},{1,2,3}}
{{3},{2,3},{1,2,3}}
{{1,2},{1,3},{2,3}}
{{1,2},{1,3},{1,2,3}}
{{1,2},{2,3},{1,2,3}}
{{1,3},{2,3},{1,2,3}}
{{1},{1,2},{1,3},{1,2,3}}
{{2},{1,2},{2,3},{1,2,3}}
{{3},{1,3},{2,3},{1,2,3}}
{{1,2},{1,3},{2,3},{1,2,3}}

Cf. A001206, A006126, A048143, A051185, A134958, A030019, A304985, A305844.

Length/@Table[Select[Subsets[Rest[Subsets[Range[n]]]],And[Union@@#==Range[n],FreeQ[Intersection@@@Tuples[#,2],{}]]&],{n,1,4}]

Inverse binomial transform of A051185.

A305854 Number of unlabeled spanning intersecting set-systems on n vertices.

Original entry on oeis.org

1, 1, 2, 10, 110, 14868, 1289830592

Offset: 0

Gus Wiseman, Jun 11 2018

An intersecting set-system S is a finite set of finite nonempty sets (edges), any two of which have a nonempty intersection. S is spanning if every vertex is contained in some edge.

			Non-isomorphic representatives of the a(3) = 10 spanning intersecting set-systems:
  {{1,2,3}}
  {{3},{1,2,3}}
  {{1,3},{2,3}}
  {{2,3},{1,2,3}}
  {{3},{1,3},{2,3}}
  {{3},{2,3},{1,2,3}}
  {{1,2},{1,3},{2,3}}
  {{1,3},{2,3},{1,2,3}}
  {{3},{1,3},{2,3},{1,2,3}}
  {{1,2},{1,3},{2,3},{1,2,3}}

Cf. A001206, A006126, A051185, A261006, A283877, A304998, A305843, A305844, A305855-A305857.

a(n) = A305856(n) - A305856(n-1) for n > 0. - Andrew Howroyd, Aug 12 2019

a(5) from Andrew Howroyd, Aug 12 2019

a(6) from Bert Dobbelaere, Apr 28 2025

A307249 Number of simplicial complexes with n nodes.

Original entry on oeis.org

1, 1, 2, 9, 114, 6894, 7785062, 2414627396434, 56130437209370320359966, 286386577668298410623295216696338374471993

Offset: 0

Gus Wiseman, Mar 31 2019

Except for a(0) = 1, this is also the number of antichains of nonempty sets covering n vertices (A006126). There are two antichains of size zero, namely {} and {{}}, while there is only one simplicial complex, namely {}. The unlabeled case is A261005. The non-covering case is A014466.

			Maximal simplices of the a(0) = 1 through a(3) = 9 simplicial complexes:
  {}    {{1}}  {{12}}    {{123}}
               {{1}{2}}  {{1}{23}}
                         {{2}{13}}
                         {{3}{12}}
                         {{12}{13}}
                         {{12}{23}}
                         {{13}{23}}
                         {{1}{2}{3}}
                         {{12}{13}{23}}

Francisco Ponce Carrión and Seth Sullivant, Marginal Independence and Partial Set Partitions, arXiv:2402.16292 [math.ST], 2024. See p. 21.
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, A305000, A305844, A306550, A317674, A319721, A320449.

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],{2,n}],SubsetQ]],{n,0,nn}]

Inverse binomial transform of A014466.

a(9) from Dmitry I. Ignatov, Nov 25 2023

A001206 Number of self-dual monotone Boolean functions of n variables.

Original entry on oeis.org

0, 1, 2, 4, 12, 81, 2646, 1422564, 229809982112, 423295099074735261880

Offset: 0

N. J. A. Sloane

Sometimes called Hosten-Morris numbers (or HM numbers).
Also the number of simplicial complexes on the set {1, ..., n-1} such that no pair of faces covers all of {1, ..., n-1}. [Miller-Sturmfels] - N. J. A. Sloane, Feb 18 2008
Also the maximal number of generators of a neighborly monomial ideal in n variables. [Miller-Sturmfels]. - N. J. A. Sloane, Feb 18 2008
Also the number of intersecting antichains on a labeled (n-1)-set or (n-1)-variable Boolean functions in the Post class F(7,2). Cf. A059090. - Vladeta Jovovic, Goran Kilibarda, Dec 28 2000
Also the number of nondominated coteries on n members. - Don Knuth, Sep 01 2005
The number of maximal families of intersecting subsets of an n-element set. - Bridget Tenner, Nov 16 2006
Rivière gives a(n) for n <= 5. - N. J. A. Sloane, May 12 2012

			a(2) = 1 + 1 = 2;
a(3) = 1 + 3 = 4;
a(4) = 1 + 7 + 3 + 1 = 12;
a(5) = 1 + 15 + 30 + 30 + 5 = 81;
a(6) = 1 + 31 + 195 + 605 + 780 + 543 + 300 + 135 + 45 + 10 + 1 = 2646;
a(7) = 1 + 63 + 1050 + 9030 + 41545 + 118629 + 233821 + 329205 + 327915 + 224280 + 100716 + 29337 + 5950 + 910 + 105 + 1 = 1422564.
Cf. A059090.
From _Gus Wiseman_, Jul 03 2019: (Start)
The a(1) = 1 through a(4) = 12 intersecting antichains of nonempty sets (see Jovovic and Kilibarda's comment):
  {}  {}     {}       {}
      {{1}}  {{1}}    {{1}}
             {{2}}    {{2}}
             {{1,2}}  {{3}}
                      {{1,2}}
                      {{1,3}}
                      {{2,3}}
                      {{1,2,3}}
                      {{1,2},{1,3}}
                      {{1,2},{2,3}}
                      {{1,3},{2,3}}
                      {{1,2},{1,3},{2,3}}
(End)

Martin Aigner and Günter M. Ziegler, Proofs from THE BOOK, Third Edition, Springer-Verlag, 2004. See chapter 22.
V. Jovovic and G. Kilibarda, The number of n-variable Boolean functions in the Post class F(7,2), Belgrade, 2001, in preparation.
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.
Charles F. Mills and W. M. Mills, The calculation of λ(8), preprint, 1979. Gives a(8).
E. Miller and B. Sturmfels, Combinatorial Commutative Algebra, Springer, 2005.
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).

Taras Banakh, Volodymyr Gavrylkiv, Automorphism groups of superextensions of groups, arXiv:1802.05804 [math.GR], 2018.
Jan C. Bioch and Toshihide Ibaraki, Generating and approximating nondominated coteries, IEEE Transactions on parallel and distributed systems 6 (1995), 905-914.
A. E. Brouwer and A. Verbeek, Counting families of mutually intersecting sets, Report ZN 41, March 1972, Math. Centr., Amsterdam. Gives a(n) for n <= 7.
A. E. Brouwer and A. Verbeek, Counting families of mutually intersecting sets, Electronic Journal of Combinatorics, Volume 20, Issue 2 (2013), Paper #P8.
Gábor Damásdi, Stefan Felsner, António Girão, Balázs Keszegh, David Lewis, Dániel T. Nagy, Torsten Ueckerdt, On Covering Numbers, Young Diagrams, and the Local Dimension of Posets, arXiv:2001.06367 [math.CO], 2020.
Jesús A. De Loera, Serkan Hoşten, Robert Krone, Lily Silverstein, Average Behavior of Minimal Free Resolutions of Monomial Ideals, arXiv:1802.06537 [math.AC], 2018.
Serkan Hosten and Walter D. Morris, Jr., The order dimension of the complete graph, Discrete Math. 201 (1999), pp. 133-139.
D. E. Loeb, Challenges in playing multiplayer games, in Levy and Beal, editors, Heuristic Programming in Artificial Intelligence, vol. 4, Ellis Horwood, 1994. [broken link]
D. E. Loeb and A. Meyerowitz, The maximal intersecting family of sets graph, in H. Barcelo and G. Kalai, editors, Proceedings of the Conference on Jerusalem Combinatorics 1993. AMS series Contemporary Mathematics, 1994.
Bartłomiej Pawelski, Counting and generating monotone Boolean functions, Doctoral Diss., Univ. Gdańsk, (Poland, 2024). See pp. 12, 15, 38, 49, 58, 73.
Bartlomiej Pawelski and Andrzej Szepietowski, Divisibility properties of Dedekind numbers, arXiv:2302.04615 [math.CO], 2023.
N. M. Rivière, Recursive formulas on free distributive lattices, J. Combinatorial Theory 5 1968 229--234. MR0231764 (38 #92).
Tom Trotter, An Application of the Erdős/Stone Theorem, Slides, Sept. 13, 2001.
Index entries for sequences related to Boolean functions

Cf. A000372, A059090.
The case with empty edges allowed is A326372.
The maximal case is A007363, or A326363 with empty edges allowed.
The case with empty intersection is A326366.
The inverse binomial transform is the covering case A305844.
Cf. A006126, A014466, A051185, A326361, A326362.

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}],Or[Intersection[#1,#2]=={},SubsetQ[#1,#2]]&]],{n,0,5}] (* Gus Wiseman, Jul 03 2019 *)

a(n+1) = Sum_{m=0..A037952(n)} A059090(n, m).

For n > 0, a(n) = A326372(n - 1) - 1. - Gus Wiseman, Jul 03 2019

a(8) due to C. F. Mills & W. H. Mills, 1979
a(8) from Daniel E. Loeb, Jan 04 1996
a(8) confirmed by Don Knuth, Feb 08 2008
a(9) from Andries E. Brouwer, Aug 25 2012

A046165 Number of minimal covers of n objects.

Original entry on oeis.org

1, 1, 2, 8, 49, 462, 6424, 129425, 3731508, 152424420, 8780782707, 710389021036, 80610570275140, 12815915627480695, 2855758994821922882, 892194474524889501292, 391202163933291014701953, 240943718535427829240708786, 208683398342300491409959279244

Offset: 0

Eric W. Weisstein

nonn

No edge of a minimal cover can be a subset of any other, so minimal covers are antichains, but the converse is not true. - Gus Wiseman, Jul 03 2019

a(n) is the number of undirected graphs on n nodes for which the intersection number and independence number are equal. See Proposition 2.3.7 and Theorem 2.3.3 of the Deligeorgaki et al. paper below. - Alex Markham, Oct 13 2022

			From _Gus Wiseman_, Jul 02 2019: (Start)
The a(1) = 1 through a(3) = 8 minimal covers:
  {{1}}  {{1,2}}    {{1,2,3}}
         {{1},{2}}  {{1},{2,3}}
                    {{2},{1,3}}
                    {{3},{1,2}}
                    {{1,2},{1,3}}
                    {{1,2},{2,3}}
                    {{1},{2},{3}}
                    {{1,3},{2,3}}
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..113
Damian Bursztyn, François Goasdoué, and Ioana Manolescu, Optimizing Reformulation-based Query Answering in RDF, [Research Report] RR-8646, INRIA Saclay. 2014.
D. Deligeorgaki, A. Markham, P. Misra, and L. Solus, Combinatorial and algebraic perspectives on the marginal independence structure of Bayesian networks, arXiv:2210.00822 [stat.ME], 2022.
Giovanni Resta, Illustration of a(4)=49.
Eric Weisstein's World of Mathematics, Minimal Cover

Cf. A035348, A000371, A003465.
Antichain covers are A006126.
Minimal covering simple graphs are A053530.
Maximal antichains are A326358.
Row sums of A035347 or of A282575.
Cf. A000372, A003182, A006602, A261005, A305844, A307249, A326359.

a:= n-> add(add((-1)^i* binomial(k,i) *(2^k-1-i)^n, i=0..k)/k!, k=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 19 2008

Table[Sum[Sum[Binomial[n,i]StirlingS2[i,k](2^k-k-1)^(n-i),{i,k,n}],{k,2,n}]+1,{n,1,20}] (* Geoffrey Critzer, Jun 27 2013 *)

E.g.f.: Sum_{n>=0} (exp(x)-1)^n*exp(x*(2^n-n-1))/n!. - Vladeta Jovovic, May 08 2004
a(n) = Sum_{k=1..n} Sum_{i=k..n} C(n,i)*Stirling2(i,k)*(2^k - k - 1)^(n - i). - Geoffrey Critzer, Jun 27 2013
a(n) ~ c * 2^(n^2/4 + n + 1/2) / sqrt(Pi*n), where c = JacobiTheta3(0,1/2) = EllipticTheta[3, 0, 1/2] = 2.1289368272118771586694585485449... if n is even, and c = JacobiTheta2(0,1/2) = EllipticTheta[2, 0, 1/2] = 2.1289312505130275585916134025753... if n is odd. - Vaclav Kotesovec, Mar 10 2014

a(0)=1 prepended by Alois P. Heinz, Feb 18 2017

cp's OEIS Frontend

A006126 Number of hierarchical models on n labeled factors or variables with linear terms forced. Also number of antichain covers of a labeled n-set.

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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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A305843 Number of labeled spanning intersecting set-systems on n vertices.

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A305854 Number of unlabeled spanning intersecting set-systems on n vertices.

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A307249 Number of simplicial complexes with n nodes.

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A001206 Number of self-dual monotone Boolean functions of n variables.

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A046165 Number of minimal covers of n objects.

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