A307249 -id:A307249 - 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

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

Original entry on oeis.org

1, 2, 5, 19, 167, 7580, 7828353, 2414682040997, 56130437228687557907787, 286386577668298411128469151667598498812365

Offset: 0

N. J. A. Sloane

A monotone Boolean function is an increasing functions from P(S), the set of subsets of S, to {0,1}.
The count of antichains includes the antichain consisting of only the empty set, but excludes the empty antichain.
Also counts bases of hereditary systems.
Also antichains of nonempty subsets of an n-set. The unlabeled case is A306505. The spanning case is A307249. This sequence has a similar description to A305000 except that the singletons must be disjoint from the other edges. - Gus Wiseman, Feb 20 2019
a(n) is the total number of hierarchical log-linear models on n labeled factors (categorical variables). See Wickramasinghe (2008) and Nardi and Rinaldo (2012). - Petros Hadjicostas, Apr 08 2020
From Lorenzo Sauras Altuzarra, Apr 02 2023: (Start)
a(n) is the number of labeled abstract simplicial complexes on n vertices.
A058673(n) <= a(n) <= A058891(n+1). (End)

			a(2)=5 from the antichains {{}}, {{1}}, {{2}}, {{1,2}}, {{1},{2}}.
From _Gus Wiseman_, Feb 20 2019: (Start)
The a(0) = 1 through a(3) = 19 antichains:
  {{}}  {{}}   {{}}      {{}}
        {{1}}  {{1}}     {{1}}
               {{2}}     {{2}}
               {{12}}    {{3}}
               {{1}{2}}  {{12}}
                         {{13}}
                         {{23}}
                         {{123}}
                         {{1}{2}}
                         {{1}{3}}
                         {{2}{3}}
                         {{1}{23}}
                         {{2}{13}}
                         {{3}{12}}
                         {{12}{13}}
                         {{12}{23}}
                         {{13}{23}}
                         {{1}{2}{3}}
                         {{12}{13}{23}}
(End)
From _Lorenzo Sauras Altuzarra_, Apr 02 2023: (Start)
The 19 sets E such that ({1, 2, 3}, E) is an abstract simplicial complex:
  {}
  {{1}}
  {{2}}
  {{3}}
  {{1}, {2}}
  {{1}, {3}}
  {{2}, {3}}
  {{1}, {2}, {3}}
  {{1}, {2}, {1, 2}}
  {{1}, {3}, {1, 3}}
  {{2}, {3}, {2, 3}}
  {{1}, {2}, {3}, {1, 2}}
  {{1}, {2}, {3}, {1, 3}}
  {{1}, {2}, {3}, {2, 3}}
  {{1}, {2}, {3}, {1, 2}, {1, 3}}
  {{1}, {2}, {3}, {1, 2}, {2, 3}}
  {{1}, {2}, {3}, {1, 3}, {2, 3}}
  {{1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}}
  {{1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}
(End)

I. Anderson, Combinatorics of Finite Sets. Oxford Univ. Press, 1987, p. 38.
Jorge Luis Arocha, "Antichains in ordered sets" [ In Spanish ]. Anales del Instituto de Matematicas de la Universidad Nacional Autonoma de Mexico 27: 1-21 (1987).
J. Berman, "Free spectra of 3-element algebras," in R. S. Freese and O. C. Garcia, editors, Universal Algebra and Lattice Theory (Puebla, 1982), Lect. Notes Math. Vol. 1004, 1983.
G. Birkhoff, Lattice Theory. American Mathematical Society, Colloquium Publications, Vol. 25, 3rd ed., Providence, RI, 1967, p. 63.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 273.
J. Dezert, Fondations pour une nouvelle théorie du raisonnement plausible et paradoxal (la DSmT), Tech. Rep. 1/06769 DTIM, ONERA, Paris, page 33, January 2003.
J. Dezert, F. Smarandache, On the generating of hyper-powersets for the DSmT, Proceedings of the 6th International Conference on Information Fusion, Cairns, Australia, 2003.
M. A. Harrison, Introduction to Switching and Automata Theory. McGraw Hill, NY, 1965, p. 188.
W. F. Lunnon, The IU function: the size of a free distributive lattice, pp. 173-181 of D. J. A. Welsh, editor, Combinatorial Mathematics and Its Applications. Academic Press, NY, 1971.
S. Muroga, Threshold Logic and Its Applications. Wiley, NY, 1971, p. 38 and 214.
D. B. West, Introduction to Graph Theory, 2nd ed., Prentice-Hall, NJ, 2001, p. 349.

Guillermo Alesandroni and Sungjin Kim, The probability that the Taylor resolution of a monomial ideal is minimal, J. Comm. Alg., 2024. See p. 3.
K. Atanassov, On Some of Smarandache's Problems.
K. S. Brown, Dedekind's problem.
Winfried Bruns, Pedro A. García-Sánchez, and Luca Moci, The monoid of monotone functions on a poset and arithmetic multiplicities for uniform matroids, arXiv:1902.00864 [math.CO], 2019.
Donald E. Campbell, Jack Graver, and Jerry S. Kelly, There are more strategy-proof procedures than you think, Mathematical Social Sciences 64 (2012) 263-265. - _N. J. A. Sloane_, Oct 23 2012
David Carey and Mordechai Katzman, Stanley-Reisner Ideals with Pure Resolutions, arXiv:2409.05481 [math.AC], 2024. See p. 52.
Fan Cheng, Optimality of routing on the wiretap network with simple network topology, Information Theory (ISIT), 2014 IEEE International Symposium on, June 29 2014-July 4 2014 Page(s): 786 - 790 INSPEC Accession Number: 14524545 Honolulu, HI.
Fan Cheng and Vincent Y. F. Tan, A Numerical Study on the Wiretap Network with a Simple Network Topology, arXiv preprint arXiv:1505.02862 [cs.IT], 2015.
Jean Dezert, Foundations for a new theory for plausible and paradoxical reasoning, Tech. Rep. DTIM/IED, ONERA, Paris, pp. 14-15, 2002.
Jean Dezert, Combination of paradoxical sources of information within the neutrosophic framework, Proceedings of the First International Conference on Neutrosophics (2001).
J. L. King, Brick tiling and monotone Boolean functions.
J. L. King, A change-of-coordinates from Geometry to Algebra applied to brick tilings, arXiv:math/9809176 [math.CO], 1998.
D. J. Kleitman, On Dedekind's problem: The number of monotone Boolean functions, Proc. Amer. Math. Soc. 21 (1969), 677-682.
D. J. Kleitman and G. Markowsky, On Dedekind's problem: the number of isotone Boolean functions. II, Trans. Amer. Math. Soc. 213 (1975), 373-390.
Y. Nardi and A. Rinaldo, The log-linear group-lasso estimator and its asymptotic properties, Bernoulli 18(3) (2012), 945-974; see Table 2 on p. 954.
Eric Weisstein's World of Mathematics, Antichain.
R. I. P. Wickramasinghe, Topics in log-linear models, Master of Science thesis in Statistics, Texas Tech University, Lubbock, TX, 2008. [For n = 2, the a(2) = 5 hierarchical log-linear models on two factors X and Y appear on p. 18. For n = 3, the a(3) = 19 hierarchical log-linear models on three factors X, Y, and Z, appear on p. 36. - _Petros Hadjicostas_, Apr 08 2020]
D. H. Wiedemann, A computation of the eighth Dedekind number, Order 8 (1991) 5-6.
Wikipedia, Abstract simplicial complex.
Gus Wiseman, Sequences enumerating clutters, antichains, hypertrees, and hyperforests, organized by labeling, spanning, and allowance of singletons.
Index entries for sequences related to Boolean functions

Equals A000372 - 1 = A007153 + 1.

Cf. A003182, A005465, A006126, A006602, A058673 (labeled matroids), A058891 (labeled hypergraphs), A261005, A293606, A304996, A305000, A306505, A307249, A317674, A319721, A320449, A321679.

nn=5;
stableSets[u_,Q_]:=If[Length[u]===0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r===w||Q[r,w]||Q[w,r]],Q]]]];
Table[Length[stableSets[Subsets[Range[n],{1,n}],SubsetQ]],{n,0,nn}] (* Gus Wiseman, Feb 20 2019 *)
A[s_Integer] := With[{s6 = StringPadLeft[ToString[s], 6, "0"]}, Cases[ Import["https://oeis.org/A" <> s6 <> "/b" <> s6 <> ".txt", "Table"], {, }][[All, 2]]];
A@372 - 1 (* Jean-François Alcover, Jan 07 2020 *)

Binomial transform of A307249 (or A006126 if its zeroth term is 1). - Gus Wiseman, Feb 20 2019

a(n) >= A005465(n) (because the hierarchical log-linear models on n factors always include all the conditional independence models considered by I. J. Good in A005465). - Petros Hadjicostas, Apr 24 2020

Last term from D. H. Wiedemann, personal communication.
Additional comments from Michael Somos, Jun 10 2002
Term a(9) (using A000372) from Joerg Arndt, Apr 07 2023

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

A305001 Number of labeled antichains of finite sets spanning n vertices without singletons.

Original entry on oeis.org

1, 0, 1, 5, 87, 6398, 7745253, 2414573042063, 56130437190053518791691, 286386577668298410118121281898931424413687

Offset: 0

Gus Wiseman, May 23 2018

nonn

From Gus Wiseman, Jul 03 2019: (Start)
Also the number of antichains covering n vertices and having empty intersection (meaning there is no vertex in common to all the edges). For example, the a(3) = 5 antichains are:
  {{3},{1,2}}
  {{2},{1,3}}
  {{1},{2,3}}
  {{1},{2},{3}}
  {{1,2},{1,3},{2,3}}
(End)

			The a(3) = 5 antichains:
  {{1,2,3}}
  {{1,2},{1,3}}
  {{1,2},{2,3}}
  {{1,3},{2,3}}
  {{1,2},{1,3},{2,3}}

The binomial transform is the non-covering case A307249.
The second binomial transform is A014466.
Cf. A000372, A003182, A006126, A006602, A046165, A261005, A304996, A304997, A304998, A304999, A305000, A326358, A326359.

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}],SubsetQ],And[Union@@#==Range[n],#=={}||Intersection@@#=={}]&]],{n,0,5}] (* Gus Wiseman, Jul 03 2019 *)

a(9) from A307249 - Dmitry I. Ignatov, Nov 27 2023

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

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

A326881 Number of set-systems with {} that are closed under intersection and cover n vertices.

Original entry on oeis.org

1, 1, 5, 71, 4223, 2725521, 151914530499, 28175294344381108057

Offset: 0

Gus Wiseman, Jul 30 2019

			The a(2) = 5 set-systems:
  {{},{1,2}}
  {{},{1},{2}}
  {{},{1},{1,2}}
  {{},{2},{1,2}}
  {{},{1},{2},{1,2}}

The case also closed under union is A000798.
The connected case (i.e., with maximum) is A102894.
The same for union instead of intersection is (also) A102894.
The non-covering case is A102895.
The BII-numbers of these set-systems (without the empty set) are A326880.
The unlabeled case is A326883.
Cf. A003465, A014466, A102896, A102897, A193674, A193675, A306445, A307249, A326878.

Table[Length[Select[Subsets[Subsets[Range[n]]],MemberQ[#,{}]&&Union@@#==Range[n]&&SubsetQ[#,Intersection@@@Tuples[#,2]]&]],{n,0,3}]

Inverse binomial transform of A102895. - Andrew Howroyd, Aug 10 2019

a(5)-a(7) from Andrew Howroyd, Aug 10 2019

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

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

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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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A305001 Number of labeled antichains of finite sets spanning n vertices without singletons.

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A046165 Number of minimal covers of n objects.

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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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