A007363 -id:A007363 - cp's OEIS frontend

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

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

Offset: 0

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

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

A326358 Number of maximal antichains of subsets of {1..n}.

Original entry on oeis.org

1, 2, 3, 7, 29, 376, 31746, 123805914

Offset: 0

Gus Wiseman, Jul 01 2019

A set system (set of sets) is an antichain if no element is a subset of any other.

			The a(0) = 1 through a(3) = 7 maximal antichains:
  {}  {}   {}      {}
      {1}  {12}    {123}
           {1}{2}  {1}{23}
                   {2}{13}
                   {3}{12}
                   {1}{2}{3}
                   {12}{13}{23}

Denis Bouyssou, Thierry Marchant, and Marc Pirlot, ELECTRE TRI-nB, pseudo-disjunctive: axiomatic and combinatorial results, arXiv:2410.18443 [cs.DM], 2024. See p. 14.
Dmitry I. Ignatov, A Note on the Number of (Maximal) Antichains in the Lattice of Set Partitions. In: Ojeda-Aciego, M., Sauerwald, K., Jäschke, R. (eds) Graph-Based Representation and Reasoning. ICCS 2023. Lecture Notes in Computer Science(). Springer, Cham.
Dmitry I. Ignatov, On the Number of Maximal Antichains in Boolean Lattices for n up to 7. Lobachevskii J. Math., 44 (2023), 137-146.

Antichains of sets are A000372.
Minimal covering antichains are A046165.
Maximal intersecting antichains are A007363.
Maximal antichains of nonempty sets are A326359.
Cf. A003182, A006126, A006602, A014466, A058891, A261005, A305000, A305001, A305844, A326360, A326361, A326362, A326363.

LoadPackage("grape");
      maxachP:=function(n) local g,G;
       g:=Graph(Group(()), Combinations([1..n]), function(x, g) return x; end,
          function(x, y) return not IsSubset(x, y) and not IsSubset(y, x); end, true);
       G:=AutGroupGraph(g);
       return Sum(CompleteSubgraphs(NewGroupGraph(G, g), -1, 2),
              function(c) return Length(Orbit(G, c, OnSets)); end);
     end;
       List([0..7],maxachP); # Mamuka Jibladze, Jan 26 2021

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]]]];
fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
Table[Length[fasmax[stableSets[Subsets[Range[n]],SubsetQ]]],{n,0,5}]
(* alternatively *)
maxachP[n_]:=FindIndependentVertexSet[
  Flatten[Map[Function[s, Map[# \[DirectedEdge] s &, Most[Subsets[s]]]],
    Subsets[Range[n]]]], Infinity, All];
Table[Length[maxachP[n]],{n,0,6}] (* Mamuka Jibladze, Jan 25 2021 *)

For n > 0, a(n) = A326359(n) + 1.

a(6)-a(7) from Mamuka Jibladze, Jan 26 2021

A326363 Number of maximal intersecting antichains of subsets of {1..n}.

Original entry on oeis.org

1, 2, 4, 6, 21, 169, 11749, 12160648

Offset: 0

Gus Wiseman, Jul 01 2019

A set system (set of sets) is an antichain if no element is a subset of any other, and is intersecting if no two element are disjoint.

			The a(1) = 1 through a(4) = 21 maximal intersecting antichains:
  {}   {}    {}            {}
  {1}  {1}   {1}           {1}
       {2}   {2}           {2}
       {12}  {3}           {3}
             {123}         {4}
             {12}{13}{23}  {1234}
                           {12}{13}{23}
                           {12}{14}{24}
                           {13}{14}{34}
                           {23}{24}{34}
                           {12}{134}{234}
                           {13}{124}{234}
                           {14}{123}{234}
                           {23}{124}{134}
                           {24}{123}{134}
                           {34}{123}{124}
                           {12}{13}{14}{234}
                           {12}{23}{24}{134}
                           {13}{23}{34}{124}
                           {14}{24}{34}{123}
                           {123}{124}{134}{234}

The case with nonempty, non-singleton edges is A326362.
Antichains of nonempty, non-singleton sets are A307249.
Minimal covering antichains are A046165.
Maximal intersecting antichains are A007363.
Maximal antichains of nonempty sets are A326359.
Cf. A000372, A003182, A006126, A006602, A014466, A051185, A058891, A261005, A305000, A305844, A326358, A326360, A326361.

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]]]];
fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
Table[Length[fasmax[stableSets[Subsets[Range[n],{0,n}],Or[Intersection[#1,#2]=={},SubsetQ[#1,#2]]&]]],{n,0,5}]
(* 2nd program *)
n = 2^6; g = CompleteGraph[n]; i = 0;
While[i < n, i++; j = i; While[j < n, j++; If[BitAnd[i, j] == 0 || BitAnd[i, j] == i || BitAnd[i, j] == j, g = EdgeDelete[g, i <-> j]]]];
sets = FindClique[g, Infinity, All];
Length[sets] (* Elijah Beregovsky, May 06 2020 *)

For n > 1, a(n) = A007363(n + 1) + 1 = A326362(n) + n + 1.

a(7) from Elijah Beregovsky, May 06 2020

A326359 Number of maximal antichains of nonempty subsets of {1..n}.

Original entry on oeis.org

1, 1, 2, 6, 28, 375, 31745, 123805913

Offset: 0

Gus Wiseman, Jul 01 2019

A set system (set of sets) is an antichain if no element is a subset of any other.

			The a(0) = 1 through a(4) = 28 antichains:
  {}   {1}    {12}      {123}           {1234}
              {1}{2}    {1}{23}         {1}{234}
                        {2}{13}         {2}{134}
                        {3}{12}         {3}{124}
                        {1}{2}{3}       {4}{123}
                        {12}{13}{23}    {1}{2}{34}
                                        {1}{3}{24}
                                        {1}{4}{23}
                                        {2}{3}{14}
                                        {2}{4}{13}
                                        {3}{4}{12}
                                        {1}{2}{3}{4}
                                        {12}{134}{234}
                                        {13}{124}{234}
                                        {14}{123}{234}
                                        {23}{124}{134}
                                        {24}{123}{134}
                                        {34}{123}{124}
                                        {1}{23}{24}{34}
                                        {2}{13}{14}{34}
                                        {3}{12}{14}{24}
                                        {4}{12}{13}{23}
                                        {12}{13}{14}{234}
                                        {12}{23}{24}{134}
                                        {13}{23}{34}{124}
                                        {14}{24}{34}{123}
                                        {123}{124}{134}{234}
                                        {12}{13}{14}{23}{24}{34}

Dmitry I. Ignatov, On the Number of Maximal Antichains in Boolean Lattices for n up to 7. Lobachevskii J. Math., 44 (2023), 137-146.

Antichains of nonempty sets are A014466.
Minimal covering antichains are A046165.
Maximal intersecting antichains are A007363.
Maximal antichains of sets are A326358.
Cf. A000372, A003182, A006126, A006602, A014466, A058891, A261005, A305000, A305844, A307249, A326360, A326362, A326363.

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]]]];
fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
Table[Length[fasmax[stableSets[Subsets[Range[n],{1,n}],SubsetQ]]],{n,0,5}]

For n > 0, a(n) = A326358(n) - 1.

a(6) from Andrew Howroyd, Aug 14 2019

a(7) from Dmitry I. Ignatov, Oct 12 2021

A326361 Number of maximal intersecting antichains of sets covering n vertices with no singletons.

Original entry on oeis.org

1, 1, 1, 2, 12, 133, 11386, 12143511

Offset: 0

Gus Wiseman, Jul 01 2019

Covering means there are no isolated vertices. A set system (set of sets) is an antichain if no part is a subset of any other, and is intersecting if no two parts are disjoint.

			The a(4) = 12 antichains:
  {{1,2,3,4}}
  {{1,2},{1,3,4},{2,3,4}}
  {{1,3},{1,2,4},{2,3,4}}
  {{1,4},{1,2,3},{2,3,4}}
  {{2,3},{1,2,4},{1,3,4}}
  {{2,4},{1,2,3},{1,3,4}}
  {{3,4},{1,2,3},{1,2,4}}
  {{1,2},{1,3},{1,4},{2,3,4}}
  {{1,2},{2,3},{2,4},{1,3,4}}
  {{1,3},{2,3},{3,4},{1,2,4}}
  {{1,4},{2,4},{3,4},{1,2,3}}
  {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}

Antichains of nonempty, non-singleton sets are A307249.
Minimal covering antichains are A046165.
Maximal intersecting antichains are A007363.
Maximal antichains of nonempty sets are A326359.
Cf. A000372, A003182, A006126, A006602, A014466, A051185, A058891, A261005, A305000, A305844, A326358, A326360, A326362, A326363.

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]]]];
fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
Table[Length[fasmax[Select[stableSets[Subsets[Range[n]],Or[Intersection[#1,#2]=={},SubsetQ[#1,#2]]&],Union@@#==Range[n]&]]],{n,0,5}]
(* 2nd program *)
n = 2^6; g = CompleteGraph[n]; i = 0;
While[i < n, i++; j = i; While[j < n, j++; If[BitAnd[i, j] == 0 || BitAnd[i, j] == i || BitAnd[i, j] == j, g = EdgeDelete[g, i <-> j]]]];
sets = Select[FindClique[g, Infinity, All], BitOr @@ # == n - 1 &];
Length[sets] (* Elijah Beregovsky, May 05 2020 *)

a(6)-a(7) from Elijah Beregovsky, May 05 2020

A326362 Number of maximal intersecting antichains of nonempty, non-singleton subsets of {1..n}.

Original entry on oeis.org

1, 1, 1, 2, 16, 163, 11742, 12160640

Offset: 0

Gus Wiseman, Jul 01 2019

A set system (set of sets) is an antichain if no part is a subset of any other, and is intersecting if no two parts are disjoint.

			The a(4) = 16 maximal intersecting antichains:
  {{1,2,3,4}}
  {{1,2},{1,3},{2,3}}
  {{1,2},{1,4},{2,4}}
  {{1,3},{1,4},{3,4}}
  {{2,3},{2,4},{3,4}}
  {{1,2},{1,3,4},{2,3,4}}
  {{1,3},{1,2,4},{2,3,4}}
  {{1,4},{1,2,3},{2,3,4}}
  {{2,3},{1,2,4},{1,3,4}}
  {{2,4},{1,2,3},{1,3,4}}
  {{3,4},{1,2,3},{1,2,4}}
  {{1,2},{1,3},{1,4},{2,3,4}}
  {{1,2},{2,3},{2,4},{1,3,4}}
  {{1,3},{2,3},{3,4},{1,2,4}}
  {{1,4},{2,4},{3,4},{1,2,3}}
  {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}

Antichains of nonempty, non-singleton sets are A307249.
Minimal covering antichains are A046165.
Maximal intersecting antichains are A007363.
Maximal antichains of nonempty sets are A326359.
Cf. A000372, A003182, A006126, A006602, A014466, A051185, A058891, A261005, A305000, A305844, A326358, A326360, A326361, A326363.

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]]]];
fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
Table[Length[fasmax[stableSets[Subsets[Range[n],{2,n}],Or[Intersection[#1,#2]=={},SubsetQ[#1,#2]]&]]],{n,0,5}]
(* 2nd program *)
n = 2^6; g = CompleteGraph[n]; i = 0;
While[i < n, i++; j = i; While[j < n, j++; If[BitAnd[i, j] == 0 || BitAnd[i, j] == i || BitAnd[i, j] == j, g = EdgeDelete[g, i <-> j]]]];
sets = FindClique[g, Infinity, All];
Length[sets]-Log[2,n]-1 (* Elijah Beregovsky, May 06 2020 *)

For n > 1, a(n) = A326363(n) - n - 1 = A007363(n + 1) - n.

a(7) from Elijah Beregovsky, May 06 2020

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

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Feb 20 2019

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

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

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

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

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

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

Partial sums of A006602 minus 1.

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

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

A326360 Number of maximal antichains of nonempty, non-singleton subsets of {1..n}.

Original entry on oeis.org

1, 1, 1, 2, 13, 279, 29820, 123590767

Offset: 0

Gus Wiseman, Jul 01 2019

A set system (set of sets) is an antichain if no element is a subset of any other.

			The a(1) = 1 through a(4) = 13 maximal antichains:
  {}  {12}  {123}         {1234}
            {12}{13}{23}  {12}{134}{234}
                          {13}{124}{234}
                          {14}{123}{234}
                          {23}{124}{134}
                          {24}{123}{134}
                          {34}{123}{124}
                          {12}{13}{14}{234}
                          {12}{23}{24}{134}
                          {13}{23}{34}{124}
                          {14}{24}{34}{123}
                          {123}{124}{134}{234}
                          {12}{13}{14}{23}{24}{34}

Dmitry I. Ignatov, On the Number of Maximal Antichains in Boolean Lattices for n up to 7. Lobachevskii J. Math., 44 (2023), 137-146.
Dmitry I. Ignatov, Supporting iPython code and input files for a(7) based on inequivalent maximal antichains for n=7 and related sequences, Github repository, section 3
Dmitry I. Ignatov, PDF version of the supporting iPython notebook for a(7)
Dmitry I. Ignatov, Supporting iPython notebook for a(7): A326360.ipynb

Antichains of nonempty, non-singleton sets are A307249.
Minimal covering antichains are A046165.
Maximal intersecting antichains are A007363.
Maximal antichains of nonempty sets are A326359.
Cf. A000372, A003182, A006126, A006602, A014466, A058891, A261005, A305000, A305844, A326358, A326361, A326362, A326363.

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]]]];
fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
Table[Length[fasmax[stableSets[Subsets[Range[n],{2,n}],SubsetQ]]],{n,0,4}]

# see Ignatov links
# Dmitry I. Ignatov, Oct 14 2021

a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*A326359(k) for n >= 2. - Andrew Howroyd, Nov 19 2021

a(6) from Andrew Howroyd, Aug 14 2019

a(7) from Dmitry I. Ignatov, Oct 14 2021

A326365 Number of intersecting antichains with empty intersection (meaning there is no vertex in common to all the edges) covering n vertices.

Original entry on oeis.org

1, 0, 0, 1, 23, 1834, 1367903, 229745722873, 423295077919493525420

Offset: 0

Gus Wiseman, Jul 01 2019

Covering means there are no isolated vertices. A set system (set of sets) is an antichain if no part is a subset of any other, and is intersecting if no two parts are disjoint.

			The a(4) = 23 intersecting antichains with empty intersection:
  {{1,2},{1,3},{2,3,4}}
  {{1,2},{1,4},{2,3,4}}
  {{1,2},{2,3},{1,3,4}}
  {{1,2},{2,4},{1,3,4}}
  {{1,3},{1,4},{2,3,4}}
  {{1,3},{2,3},{1,2,4}}
  {{1,3},{3,4},{1,2,4}}
  {{1,4},{2,4},{1,2,3}}
  {{1,4},{3,4},{1,2,3}}
  {{2,3},{2,4},{1,3,4}}
  {{2,3},{3,4},{1,2,4}}
  {{2,4},{3,4},{1,2,3}}
  {{1,2},{1,3,4},{2,3,4}}
  {{1,3},{1,2,4},{2,3,4}}
  {{1,4},{1,2,3},{2,3,4}}
  {{2,3},{1,2,4},{1,3,4}}
  {{2,4},{1,2,3},{1,3,4}}
  {{3,4},{1,2,3},{1,2,4}}
  {{1,2},{1,3},{1,4},{2,3,4}}
  {{1,2},{2,3},{2,4},{1,3,4}}
  {{1,3},{2,3},{3,4},{1,2,4}}
  {{1,4},{2,4},{3,4},{1,2,3}}
  {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}

Intersecting antichain covers are A305844.
Intersecting covers with empty intersection are A326364.
Antichain covers with empty intersection are A305001.
The binomial transform is the non-covering case A326366.
Covering, intersecting antichains with empty intersection are A326365.
Cf. A006126, A007363, A014466, A051185, A058891, A305843, A307249, A318128, A318129, A326361, A326362, A326363.

stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],Or[Intersection[#1,#2]=={},SubsetQ[#1,#2]]&],And[Union@@#==Range[n],#=={}||Intersection@@#=={}]&]],{n,0,4}]

a(7)-a(8) from Andrew Howroyd, Aug 14 2019

cp's OEIS Frontend

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

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

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A326358 Number of maximal antichains of subsets of {1..n}.

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A326363 Number of maximal intersecting antichains of subsets of {1..n}.

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A326359 Number of maximal antichains of nonempty subsets of {1..n}.

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A326361 Number of maximal intersecting antichains of sets covering n vertices with no singletons.

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A326362 Number of maximal intersecting antichains of nonempty, non-singleton subsets of {1..n}.

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