A000372 -id:A000372 - cp's OEIS frontend

A367769 Number of finite sets of nonempty non-singleton subsets of {1..n} contradicting a strict version of the axiom of choice.

0, 0, 0, 1, 1490, 67027582, 144115188036455750, 1329227995784915872903806998967001298, 226156424291633194186662080095093570025917938800079226639565284090686126876

Offset: 0

Gus Wiseman, Dec 05 2023

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

Includes all set-systems with more edges than covered vertices, but this condition is not sufficient.

			The a(3) = 1 set-system is: {{1,2},{1,3},{2,3},{1,2,3}}.

Wikipedia, Axiom of choice.

Set-systems without singletons are counted by A016031, covering A323816.
The complement is A367770, with singletons allowed A367902 (ranks A367906).
The version for simple graphs is A367867, covering A367868.
The version allowing singletons and empty edges is A367901.
The version allowing singletons is A367903, ranks A367907.
A000372 counts antichains, covering A006126, nonempty A014466.
A003465 counts covering set-systems, unlabeled A055621.
A058891 counts set-systems, unlabeled A000612.
A059201 counts covering T_0 set-systems.
A323818 counts covering connected set-systems.
Cf. A007716, A092918, A102896, A283877, A306445, A326031, A355739, A355740, A355741, A367904, A367905.

Table[Length[Select[Subsets[Select[Subsets[Range[n]], Length[#]>1&]], Select[Tuples[#], UnsameQ@@#&]=={}&]], {n,0,3}]

a(n) = 2^(2^n-n-1) - A367770(n) = A016031(n+1) - A367770(n). - Christian Sievers, Jul 28 2024

a(6)-a(8) from Christian Sievers, Jul 28 2024

A056941 Number of antichains (or order ideals) in the poset 5mn or plane partitions with not more than m rows, n columns and entries <= 5.

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 21, 21, 1, 1, 56, 196, 56, 1, 1, 126, 1176, 1176, 126, 1, 1, 252, 5292, 14112, 5292, 252, 1, 1, 462, 19404, 116424, 116424, 19404, 462, 1, 1, 792, 60984, 731808, 1646568, 731808, 60984, 792, 1, 1, 1287, 169884, 3737448, 16818516, 16818516, 3737448, 169884, 1287, 1

Offset: 0

Mitch Harris

Triangle of generalized binomial coefficients (n,k)A342889.%20-%20_N.%20J.%20A.%20Sloane">5; cf. A342889. - _N. J. A. Sloane, Apr 03 2021

			The array starts:
  [1    1      1        1          1           1            1 ...]
  [1    6     21       56        126         252          462 ...]
  [1   21    196     1176       5292       19404        60984 ...]
  [1   56   1176    14112     116424      731808      3737448 ...]
  [1  126   5292   116424    1646568    16818516    133613766 ...]
  [1  252  19404   731808   16818516   267227532   3184461423 ...]
  [1  462  60984  3737448  133613766  3184461423  55197331332 ...]
  [...]
Considered as a triangle, the initial rows are:
   1;
   1,   1;
   1,   6,     1;
   1,  21,    21,      1;
   1,  56,   196,     56,       1;
   1, 126,  1176,   1176,     126,      1;
   1, 252,  5292,  14112,    5292,    252,     1;
   1, 462, 19404, 116424,  116424,  19404,   462,   1;
   1, 792, 60984, 731808, 1646568, 731808, 60984, 792, 1; ...

Berman and Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), p. 103-124
P. A. MacMahon, Combinatory Analysis, Section 495, 1916.
R. P. Stanley, Theory and application of plane partitions. II. Studies in Appl. Math. 50 (1971), p. 259-279. Thm. 18.1

Seiichi Manyama, Rows n = 0..139 of triangle, flattened
J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124. [Annotated scanned copy]
Johann Cigler, Pascal triangle, Hoggatt matrices, and analogous constructions, arXiv:2103.01652 [math.CO], 2021.
Johann Cigler, Some observations about Hoggatt triangles, Universität Wien (Austria, 2021).
P. A. MacMahon, Combinatory analysis.
Index entries for sequences related to posets

Cf. A000372, A001263, A005363 (row sums), A056932, A056939, A056940, A142465, A142467, A142468.
Antidiagonals sum to A005363 (Hoggatt sequence).
Triangles of generalized binomial coefficients (n,k)_m (or generalized Pascal triangles) for m = 1,...,12: A007318 (Pascal), A001263, A056939, A056940, A056941, A142465, A142467, A142468, A174109, A342889, A342890, A342891.

A056941:= func< n,k | (&*[Binomial(n+j,k)/Binomial(k+j,k): j in [0..4]]) >;
[A056941(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 14 2022

T[n_, k_] := Product[Binomial[n+j, k]/Binomial[k+j, k], {j,0,4}];
Table[T[n, k], {n,0,13}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 14 2022 *)

A056941(n,m)=prod(k=0,4,binomial(n+m+k,m+k)/binomial(n+k,k)) \\ as an array \\ M. F. Hasler, Sep 26 2018

def A056941(n,k): return product(binomial(n+j,k)/binomial(k+j,k) for j in (0..4))
flatten([[A056941(n,k) for k in range(n+1)] for n in range(14)]) # G. C. Greubel, Nov 14 2022

From Peter Bala, Oct 13 2011: (Start)
A(n, k) = Product_{j=0..4} C(n+k+j, k+j)/C(n+j, j) gives the array as a square.
g(n-1, k-1)*g(n, k+1)*g(n+1, k) = g(n-1, k)*g(n, k-1)*g(n+1, k+1) where g(n, k) is the array A(n, k) and triangle T(n, k).
Define f(r,n) = n!*(n+1)!*...*(n+r)!. The triangle whose (n,k)-th entry is f(r,0)*f(r,n)/(f(r,k)*f(r,n-k)) is A007318 (r = 0), A001263 (r = 1), A056939 (r = 2), A056940 (r = 3) and A056941 (r = 4). (End)
From Peter Bala, May 10 2012: (Start)
Determinants of 5 X 5 subarrays of Pascal's triangle A007318 (a matrix entry being set to 0 when not present).
Also determinants of 5 X 5 arrays whose entries come from a single row:
  det [C(n,k), C(n,k-1), C(n,k-2), C(n,k-3), C(n,k-4); C(n,k+1), C(n,k), C(n,k-1), C(n,k-2), C(n,k-3); C(n,k+2), C(n,k+1), C(n,k), C(n,k-1), C(n,k-2); C(n,k+3), C(n,k+2), C(n,k+1), C(n,k), C(n,k-1); C(n,k+4), C(n,k+3), C(n,k+2), C(n,k+1), C(n,k)]. (End)
From G. C. Greubel, Nov 14 2022: (Start)
T(n, k) = Product_{j=0..4} binomial(n+j, k)/binomial(k+j, k) (gives the triangle).
Sum_{k=0..n} T(n, k) = A005363(n). (End)

Edited by M. F. Hasler, Sep 26 2018

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

A367770 Number of sets of nonempty non-singleton subsets of {1..n} satisfying a strict version of the axiom of choice.

Original entry on oeis.org

1, 1, 2, 15, 558, 81282, 39400122, 61313343278, 309674769204452

Offset: 0

Gus Wiseman, Dec 05 2023

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

Excludes all set-systems with more edges than covered vertices, but this condition is not sufficient.

			The a(3) = 15 set-systems:
  {}
  {{1,2}}
  {{1,3}}
  {{2,3}}
  {{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}}
  {{1,2},{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}}

Wikipedia, Axiom of choice.

Set-systems without singletons are counted by A016031, covering A323816.
The version for simple graphs is A133686, covering A367869.
The complement is counted by A367769.
The complement allowing singletons and empty sets is A367901.
Allowing singletons gives A367902, ranks A367906.
The complement allowing singletons is A367903, ranks A367907.
These set-systems have ranks A367906 /\ A326781.
A000372 counts antichains, covering A006126, nonempty A014466.
A003465 counts covering set-systems, unlabeled A055621.
A058891 counts set-systems, unlabeled A000612.
A323818 counts covering connected set-systems, unlabeled A323819.
Cf. A059201, A083323, A092918, A102896, A283877, A305000, A306445, A355739, A355740, A367904, A367905.

Table[Length[Select[Subsets[Select[Subsets[Range[n]], Length[#]>1&]], Select[Tuples[#], UnsameQ@@#&]!={}&]],{n,0,3}]

a(6)-a(8) from Christian Sievers, Jul 28 2024

A056940 Number of antichains (or order ideals) in the poset 4mn or plane partitions with at most m rows and n columns and entries <= 4.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 15, 15, 1, 1, 35, 105, 35, 1, 1, 70, 490, 490, 70, 1, 1, 126, 1764, 4116, 1764, 126, 1, 1, 210, 5292, 24696, 24696, 5292, 210, 1, 1, 330, 13860, 116424, 232848, 116424, 13860, 330, 1, 1, 495, 32670, 457380, 1646568, 1646568, 457380, 32670, 495, 1

Offset: 0

Mitch Harris

Triangle of generalized binomial coefficients (n,k)A342889.%20-%20_N.%20J.%20A.%20Sloane">4; cf. A342889. - _N. J. A. Sloane, Apr 03 2021
Determinants of 4 X 4 subarrays of Pascal's triangle A007318 (a matrix entry being set to 0 when not present). - Gerald McGarvey, Feb 24 2005
Row sums are: {1, 2, 7, 32, 177, 1122, 7898, 60398, 494078, 4274228, 38763298, ...}. - Roger L. Bagula, Mar 08 2010
Also determinants of 4x4 arrays whose entries come from a single row: T(n,k) = det [C(n,k), C(n,k-1), C(n,k-2), C(n,k-3); C(n,k+1), C(n,k), C(n,k-1), C(n,k-2); C(n,k+2), C(n,k+1), C(n,k), C(n,k-1); C(n,k+3), C(n,k+2), C(n,k+1), C(n,k)]. - Peter Bala, May 10 2012

			Triangle begins as:
  1.
  1,   1.
  1,   5,     1.
  1,  15,    15,      1.
  1,  35,   105,     35,      1.
  1,  70,   490,    490,     70,      1.
  1, 126,  1764,   4116,   1764,    126,     1.
  1, 210,  5292,  24696,  24696,   5292,   210,   1.
  1, 330, 13860, 116424, 232848, 116424, 13860, 330, 1. - _Roger L. Bagula_, Mar 08 2010

G. C. Greubel, Rows n = 0..100 of triangle, flattened
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124. [Annotated scanned copy]
Johann Cigler, Pascal triangle, Hoggatt matrices, and analogous constructions, arXiv:2103.01652 [math.CO], 2021.
Johann Cigler, Some observations about Hoggatt triangles, Universität Wien (Austria, 2021).
P. A. MacMahon, Combinatory analysis, sect. 495, 1916.
R. P. Stanley, Theory and application of plane partitions, II. Studies in Appl. Math. 50 (1971), p. 259-279. DOI:10.1002/sapm1971503259. Thm. 18.1.
Index entries for sequences related to posets

Cf. A000372, A056932, A001263.
Antidiagonals sum to A005362 (Hoggatt sequence).
Cf. A056939 (q=2), A056940 (q=3), A056941 (q=4), A142465 (q=5), A142467 (q=6), A142468 (q=7), A174109 (q=8).
Triangles of generalized binomial coefficients (n,k)_m (or generalized Pascal triangles) for m = 1,...,12: A007318 (Pascal), A001263, A056939, A056940, A056941, A142465, A142467, A142468, A174109, A342889, A342890, A342891.

c[n_, q_] = Product[i + j, {j, 0, q}, {i, 1, n}];
T[n_, m_, q_] = c[n, q]/(c[m, q]*c[n - m, q]);
Table[T[n, k, 3], {n, 0, 10}, {k, 0, n}]//Flatten (* Roger L. Bagula, Mar 08 2010 *)(* modified by G. C. Greubel, Apr 13 2019 *)

A056940(n,m)=prod(k=0,3,binomial(n+m+k,m+k)/binomial(n+k,k)) \\ M. F. Hasler, Sep 26 2018

Product_{k=0..3} C(n+m+k, m+k)/C(n+k, k) gives the array as a square.
T(n,m,q) = c(n,q)/(c(m,q)*c(n-m,q)) with c(n,q) = Product_{i=1..n, j=0..q} (i + j), q = 3. - Roger L. Bagula, Mar 08 2010
From Peter Bala, Oct 13 2011: (Start)
T(n-1,k-1)*T(n,k+1)*T(n+1,k) = T(n-1,k)*T(n,k-1)*T(n+1,k+1).
Define f(r,n) = n!*(n+1)!*...*(n+r)!. The triangle whose (n,k)-th entry is f(r,0)*f(r,n)/(f(r,k)*f(r,n-k)) is A007318 (r = 0), A001263 (r = 1), A056939 (r = 2), A056940 (r = 3) and A056941 (r = 4). (End)

Edited by M. F. Hasler, Sep 26 2018

A324167 Number of non-crossing antichain covers of {1,...,n}.

Original entry on oeis.org

1, 1, 2, 9, 67, 633, 6763, 77766, 938957, 11739033, 150649945, 1973059212, 26265513030, 354344889798, 4833929879517, 66568517557803, 924166526830701, 12920482325488761, 181750521972603049, 2570566932237176232, 36532394627404815308, 521439507533582646156

Offset: 0

Gus Wiseman, Feb 17 2019

nonn

An antichain is non-crossing if no pair of distinct parts is of the form {{...x...y...}, {...z...t...}} where x < z < y < t or z < x < t < y.

			The a(3) = 9 antichains:
  {{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}}

Andrew Howroyd, Table of n, a(n) for n = 0..500

Cf. A000108, A000124, A000372 (antichains), A001006, A006126 (antichain covers), A014466, A048143, A054726 (non-crossing graphs), A099947, A261005, A283877, A306438.

Cf. A324166, A324168, A324169, A324170, A324171, A324173, A359984 (no singletons).

nn=6;
croXQ[stn_]:=MatchQ[stn,{_,{_,x_,_,y_,_},_,{_,z_,_,t_,_},_}/;x

seq(n)={my(f=O(1)); for(n=2, n, f = 1 + (4*x + x^2)*f^2 - 3*x^2*(1 + x)*f^3); Vec(subst(x*(1 + x^2*f^2 - 3*x^3*f^3), x, x/(1-x))/x) } \\ Andrew Howroyd, Jan 20 2023

Inverse binomial transform of A324168.

Binomial transform of A359984. - Andrew Howroyd, Jan 20 2023

Terms a(9) and beyond from Andrew Howroyd, Jan 20 2023

A368600 Number of ways to choose a set of n nonempty subsets of {1..n} such that it is not possible to choose a different element from each.

Original entry on oeis.org

0, 0, 0, 3, 164, 18625, 5491851, 4649088885, 12219849683346

Offset: 0

Gus Wiseman, Jan 01 2024

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

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

Wikipedia, Axiom of choice.

For a unique choice we have A003024, any length A367904 (ranks A367908).
Sets of n nonempty subsets of {1..n} are counted by A136556.
For any length we have A367903, ranks A367907, no singletons A367769.
The complement is A368601, any length A367902 (see also A367770, A367906).
A000372 counts antichains, covering A006126, nonempty A014466.
A003465 counts covering set-systems, unlabeled A055621.
A058891 counts set-systems, unlabeled A000612.
A059201 counts covering T_0 set-systems.
A323818 counts covering connected set-systems, unlabeled A323819.
Cf. A003025, A088957, A133686, A334282, A355529, A355740, A367862, A367867, A367868, A367901, A368094, A368097.

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

from itertools import combinations, product, chain
from scipy.special import comb
def v(c):
    for elements in product(*c):
        if len(set(elements)) == len(elements):
            return True
    return False
def a(n):
    if n == 0:
        return 1
    subsets = list(chain.from_iterable(combinations(range(1, n + 1), r) for r in range(1, n + 1)))
    cs = combinations(subsets, n)
    c = sum(1 for c in cs if v(c))
    return c
[print(int(comb(2**n-1,n) - a(n))) for n in range(7)] # Robert P. P. McKone, Jan 02 2024

a(n) = A136556(n) - A368601(n).

a(6) from Robert P. P. McKone, Jan 02 2024

a(7)-a(8) from Christian Sievers, Jul 25 2024

A368601 Number of ways to choose a set of n nonempty subsets of {1..n} such that it is possible to choose a different element from each.

Original entry on oeis.org

1, 1, 3, 32, 1201, 151286, 62453670, 84707326890, 384641855115279

Offset: 0

Gus Wiseman, Jan 01 2024

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

			The a(2) = 3 set-systems:
  {{1},{2}}
  {{1},{1,2}}
  {{2},{1,2}}
Non-isomorphic representatives of the a(3) = 32 set-systems:
  {{1},{2},{3}}
  {{1},{2},{1,3}}
  {{1},{2},{1,2,3}}
  {{1},{1,2},{1,3}}
  {{1},{1,2},{2,3}}
  {{1},{1,2},{1,2,3}}
  {{1},{2,3},{1,2,3}}
  {{1,2},{1,3},{2,3}}
  {{1,2},{1,3},{1,2,3}}

Wikipedia, Axiom of choice.

For a unique choice we have A003024, any length A367904 (ranks A367908).
Sets of n nonempty subsets of {1..n} are counted by A136556.
For any length we have A367902, ranks A367906, no singletons A367770.
The complement is A368600, any length A367903 (see also A367907, A367769).
A000372 counts antichains, covering A006126, nonempty A014466.
A003465 counts covering set-systems, unlabeled A055621.
A058891 counts set-systems, unlabeled A000612.
A059201 counts covering T_0 set-systems.
A323818 counts covering connected set-systems, unlabeled A323819.
Cf. A003025, A088957, A133686, A334282, A355529, A355740, A367862, A367867, A367901, A367905, A368094, A368097.

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

from itertools import combinations, product, chain
def v(c):
    for elements in product(*c):
        if len(set(elements)) == len(elements):
            return True
    return False
def a(n):
    if n == 0:
        return 1
    subsets = list(chain.from_iterable(combinations(range(1, n + 1), r) for r in
range(1, n + 1)))
    cs = combinations(subsets, n)
    c = sum(1 for c in cs if v(c))
    return c
[print(a(n)) for n in range(7)] # Robert P. P. McKone, Jan 02 2024

a(n) + A368600(n) = A136556(n).

a(6) from Robert P. P. McKone, Jan 02 2024

a(7)-a(8) from Christian Sievers, Jul 25 2024

A007363 Maximal self-dual antichains on n points.

Original entry on oeis.org

0, 1, 3, 5, 20, 168, 11748, 12160647

Offset: 1

N. J. A. Sloane

From Gus Wiseman, Jul 02 2019: (Start)
If self-dual means (pairwise) intersecting, then a(n) is the number of maximal intersecting antichains of nonempty subsets of {1..(n - 1)}. A set of sets is an antichain if no part is a subset of any other, and is intersecting if no two parts are disjoint. For example, the a(2) = 1 through a(5) = 20 maximal intersecting antichains are:
   {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}
(End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Daniel E. Loeb, On Games, Voting Schemes and Distributive Lattices. LaBRI Report 625-93, University of Bordeaux I, 1993. [Broken link]

Intersecting antichains are A326372.
Intersecting antichains of nonempty sets are A001206.
Unlabeled intersecting antichains are A305857.
Maximal antichains of nonempty sets are A326359.
The case with empty edges allowed is A326363.
Cf. A000372, A305844, A306007, A326358, A326361, A326362, A326366.

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}],Or[Intersection[#1,#2]=={},SubsetQ[#1,#2]]&]]],{n,0,5}] (* Gus Wiseman, Jul 02 2019 *)
(* 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]-1 (* Elijah Beregovsky, May 06 2020 *)

For n > 0, a(n) = A326363(n - 1) - 1 = A326362(n - 1) + n - 1. - Gus Wiseman, Jul 03 2019

a(8) from Elijah Beregovsky, May 06 2020

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

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A367769 Number of finite sets of nonempty non-singleton subsets of {1..n} contradicting a strict version of the axiom of choice.

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A056941 Number of antichains (or order ideals) in the poset 5*m*n or plane partitions with not more than m rows, n columns and entries <= 5.

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

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A367770 Number of sets of nonempty non-singleton subsets of {1..n} satisfying a strict version of the axiom of choice.

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A056940 Number of antichains (or order ideals) in the poset 4*m*n or plane partitions with at most m rows and n columns and entries <= 4.

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A368600 Number of ways to choose a set of n nonempty subsets of {1..n} such that it is not possible to choose a different element from each.

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A056940 Number of antichains (or order ideals) in the poset 4mn or plane partitions with at most m rows and n columns and entries <= 4.