A293993 -id:A293993 - 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.
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Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 273.
Michael A. Harrison, Introduction to Switching and Automata Theory, McGraw Hill, NY, 1965, p. 188.
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R. A. Obando, On the number of nondegenerate monotone boolean functions of n variables in an n-variable boolean algebra. In preparation.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Douglas B. West, Introduction to Graph Theory, 2nd ed., Prentice-Hall, NJ, 2001, p. 349.

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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.
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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
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Kevin S. Brown, Generating the Monotone Boolean Functions.
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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

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

A293606 Number of unlabeled antichains of weight n.

Original entry on oeis.org

1, 1, 2, 3, 6, 9, 20, 33, 72, 139

Offset: 0

Gus Wiseman, Oct 13 2017

An antichain is a finite set of finite nonempty sets, none of which is a subset of any other. The weight of an antichain is the sum of cardinalities of its elements.
From Gus Wiseman, Aug 15 2019: (Start)
Also the number of non-isomorphic set multipartitions (multisets of sets) of weight n where every vertex is the unique common element of some subset of the edges. For example, the a(1) = 1 through a(6) = 20 set multipartitions are:
  {1}  {1}{1}  {1}{1}{1}  {1}{2}{12}    {1}{2}{2}{12}    {12}{13}{23}
       {1}{2}  {1}{2}{2}  {1}{1}{1}{1}  {1}{2}{3}{23}    {1}{2}{12}{12}
               {1}{2}{3}  {1}{1}{2}{2}  {1}{1}{1}{1}{1}  {1}{2}{13}{23}
                          {1}{2}{2}{2}  {1}{1}{2}{2}{2}  {1}{2}{3}{123}
                          {1}{2}{3}{3}  {1}{2}{2}{2}{2}  {1}{1}{2}{2}{12}
                          {1}{2}{3}{4}  {1}{2}{2}{3}{3}  {1}{1}{2}{3}{23}
                                        {1}{2}{3}{3}{3}  {1}{2}{2}{2}{12}
                                        {1}{2}{3}{4}{4}  {1}{2}{3}{3}{23}
                                        {1}{2}{3}{4}{5}  {1}{2}{3}{4}{34}
                                                         {1}{1}{1}{1}{1}{1}
                                                         {1}{1}{1}{2}{2}{2}
                                                         {1}{1}{2}{2}{2}{2}
                                                         {1}{1}{2}{2}{3}{3}
                                                         {1}{2}{2}{2}{2}{2}
                                                         {1}{2}{2}{3}{3}{3}
                                                         {1}{2}{3}{3}{3}{3}
                                                         {1}{2}{3}{3}{4}{4}
                                                         {1}{2}{3}{4}{4}{4}
                                                         {1}{2}{3}{4}{5}{5}
                                                         {1}{2}{3}{4}{5}{6}
(End)

			Non-isomorphic representatives of the a(5) = 9 antichains are:
((12345)),
((1)(2345)), ((12)(134)), ((12)(345)),
((1)(2)(345)), ((1)(23)(45)), ((2)(13)(14)),
((1)(2)(3)(45)),
((1)(2)(3)(4)(5)).

Cf. A006126, A006602, A007411, A007716, A048143, A049311, A283877, A293607.

Cf. A000372, A000612, A003182, A014466, A055621, A293993, A326704, A326972.

Euler transform of A293607.

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

A319721 Number of non-isomorphic antichains of multisets of weight n.

Original entry on oeis.org

1, 1, 4, 8, 24, 50, 148, 349, 1014, 2717, 8114

Offset: 0

Gus Wiseman, Sep 26 2018

In an antichain, no part is a proper submultiset of any other. The weight of an antichain is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(1) = 1 through a(3) = 8 antichains:
1: {{1}}
2: {{1,1}}
   {{1,2}}
   {{1},{1}}
   {{1},{2}}
3: {{1,1,1}}
   {{1,2,2}}
   {{1,2,3}}
   {{1},{2,2}}
   {{1},{2,3}}
   {{1},{1},{1}}
   {{1},{2},{2}}
   {{1},{2},{3}}

Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, Journal of Integer Sequences, Vol. 7 (2004).

Cf. A001055, A001970, A007716, A096827, A253249, A285572, A285573, A293993, A318099, A319616-A319646, A319719.

A304713 Squarefree numbers whose prime indices are pairwise indivisible. Heinz numbers of strict integer partitions with pairwise indivisible parts.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 15, 17, 19, 23, 29, 31, 33, 35, 37, 41, 43, 47, 51, 53, 55, 59, 61, 67, 69, 71, 73, 77, 79, 83, 85, 89, 91, 93, 95, 97, 101, 103, 107, 109, 113, 119, 123, 127, 131, 137, 139, 141, 143, 145, 149, 151, 155, 157, 161, 163, 165, 167, 173

Offset: 1

Gus Wiseman, May 17 2018

nonn

A prime index of n is a number m such that prime(m) divides n. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			Sequence of entries together with their corresponding multiset multisystems (see A302242) begins:
1:  {}
2:  {{}}
3:  {{1}}
5:  {{2}}
7:  {{1,1}}
11: {{3}}
13: {{1,2}}
15: {{1},{2}}
17: {{4}}
19: {{1,1,1}}
23: {{2,2}}
29: {{1,3}}
31: {{5}}
33: {{1},{3}}
35: {{2},{1,1}}

Cf. A000009, A005117, A006126, A056239, A073576, A285572, A285573, A293606, A293993, A302696, A302796, A303362, A303365, A304711.

Select[Range[300],SquareFreeQ[#]&&Select[Tuples[PrimePi/@First/@FactorInteger[#],2],UnsameQ@@#&&Divisible@@#&]==={}&]

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

A293994 Number of unlabeled multiset clutters of weight n.

Original entry on oeis.org

1, 1, 2, 3, 7, 13, 39, 88, 265

Offset: 0

Gus Wiseman, Oct 21 2017

A multiset clutter is a connected antichain of finite multisets. The weight of a multiset clutter is the sum of cardinalities (counting multiplicity) of its edges.

			Non-isomorphic representatives of the a(5) = 13 multiset clutters are:
((11111)), ((11112)), ((11122)), ((11123)), ((11223)), ((11234)), ((12345)), ((11)(122)), ((11)(123)), ((12)(111)), ((12)(113)), ((12)(134)), ((13)(122)).

Cf. A007716, A048143, A286520, A293607, A293993.

A006602 a(n) is the number of hierarchical models on n unlabeled factors or variables with linear terms forced.

Original entry on oeis.org

2, 1, 2, 5, 20, 180, 16143, 489996795, 1392195548399980210, 789204635842035039135545297410259322

Offset: 0

Colin Mallows

Also number of pure (= irreducible) group-testing histories of n items - A. Boneh, Mar 31 2000
Also number of antichain covers of an unlabeled n-set, so a(n) equals first differences of A003182. - Vladeta Jovovic, Goran Kilibarda, Aug 18 2000
Also number of inequivalent (under permutation of variables) nondegenerate monotone Boolean functions of n variables. We say h and g (functions of n variables) are equivalent if there exists a permutation p of S_n such that hp=g. E.g., a(3)=5 because xyz, xy+xz+yz, x+yz+xyz, xy+xz+xyz, x+y+z+xy+xz+yz+xyz are 5 inequivalent nondegenerate monotone Boolean functions that generate (by permutation of variables) the other 4. For example, y+xz+xyz can be obtained from x+yz+xyz by exchanging x and y. - Alan Veliz-Cuba (alanavc(AT)vt.edu), Jun 16 2006
The non-spanning/covering case is A003182. The labeled case is A006126. - Gus Wiseman, Feb 20 2019

			From _Gus Wiseman_, Feb 20 2019: (Start)
Non-isomorphic representatives of the a(0) = 2 through a(4) = 20 antichains:
  {}    {{1}}  {{12}}    {{123}}         {{1234}}
  {{}}         {{1}{2}}  {{1}{23}}       {{1}{234}}
                         {{13}{23}}      {{12}{34}}
                         {{1}{2}{3}}     {{14}{234}}
                         {{12}{13}{23}}  {{1}{2}{34}}
                                         {{134}{234}}
                                         {{1}{24}{34}}
                                         {{1}{2}{3}{4}}
                                         {{13}{24}{34}}
                                         {{14}{24}{34}}
                                         {{13}{14}{234}}
                                         {{12}{134}{234}}
                                         {{1}{23}{24}{34}}
                                         {{124}{134}{234}}
                                         {{12}{13}{24}{34}}
                                         {{14}{23}{24}{34}}
                                         {{12}{13}{14}{234}}
                                         {{123}{124}{134}{234}}
                                         {{13}{14}{23}{24}{34}}
                                         {{12}{13}{14}{23}{24}{34}}
(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 10 2020]
V. Jovovic and G. Kilibarda, On enumeration of the class of all monotone Boolean functions, in preparation.
A. A. Mcintosh, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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. 14.
V. Jovovic and G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, Diskretnaya Matematika, 11(4) (1999), 127-138 (translated in Discrete Mathematics and Applications, 9(6) (1999), 593-605).
C. Lienkaemper, When do neural codes come from convex or good covers?, 2015.
C. L. Mallows, Emails to N. J. A. Sloane, Jun-Jul 1991
Gus Wiseman, Sequences enumerating clutters, antichains, hypertrees, and hyperforests, organized by labeling, spanning, and allowance of singletons.

Cf. A000372, A003182, A006126 (labeled case), A007411, A014466, A261005, A293993, A304997, A304998, A304999, A305001, A305855, A306505, A320449, A321679.

a(n) = A007411(n) + 1.

First differences of A003182. - Gus Wiseman, Feb 23 2019

a(6) from A. Boneh, 32 Hantkeh St., Haifa 34608, Israel, Mar 31 2000
Entry revised by N. J. A. Sloane, Jul 23 2006
a(7) from A007411 and A003182. - N. J. A. Sloane, Aug 13 2015
Named edited by Petros Hadjicostas, Apr 08 2020
a(8) from A003182. - Bartlomiej Pawelski, Nov 27 2022
a(9) from A007411. - Dmitry I. Ignatov, Nov 27 2023

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

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A293606 Number of unlabeled antichains of weight n.

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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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A319721 Number of non-isomorphic antichains of multisets of weight n.

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A304713 Squarefree numbers whose prime indices are pairwise indivisible. Heinz numbers of strict integer partitions with pairwise indivisible parts.

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A307249 Number of simplicial complexes with n nodes.

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