A304997 -id:A304997 - cp's OEIS frontend

A261005 Number of unlabeled simplicial complexes with n nodes.

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

Offset: 0

N. J. A. Sloane, Aug 13 2015

nonn

Also the number of non-isomorphic antichains of nonempty sets covering n vertices. The labeled case is A006126, except with a(0) = 1. - Gus Wiseman, Feb 23 2019

			From _Gus Wiseman_, Feb 23 2019: (Start)
Non-isomorphic representatives of the a(0) = 1 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)

Benoît Jubin, Posting to Sequence Fans Mailing List, Aug 12 2015.

C. Lienkaemper, A. Shiu, and Z. Woodstock, Obstructions to convexity in neural codes, Preprint 2015.
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.

Apart from a(0), same as A006602, and after subtracting 1, A007411.

Cf. A000372, A003182, A006126, A014466, A261006, A304997, A304998, A306505, A306550, A321679.

First differences of A306505. - Gus Wiseman, Feb 23 2019

a(n) = A003182(n) - A003182(n-1) for n > 0. - Andrew Howroyd, May 28 2023

a(8)-a(9) added using A003182 by Andrew Howroyd, May 28 2023

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

A305000 Number of labeled antichains of finite sets spanning some subset of {1,...,n} with singleton edges allowed.

Original entry on oeis.org

1, 2, 8, 72, 1824, 220608, 498243968, 309072306743552, 14369391925598802012151296, 146629927766168786239127150948525247729660416

Offset: 0

Gus Wiseman, May 23 2018

Only the non-singleton edges are required to form an antichain.

Number of non-degenerate unate Boolean functions of n or fewer variables. - Aniruddha Biswas, May 11 2024

			The a(2) = 8 antichains:
  {}
  {{1}}
  {{2}}
  {{1,2}}
  {{1},{2}}
  {{1},{1,2}}
  {{2},{1,2}}
  {{1},{2},{1,2}}

Aniruddha Biswas and Palash Sarkar, Counting unate and balanced monotone Boolean functions, arXiv:2304.14069 [math.CO], 2023.
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. 4, 12.
Gus Wiseman, Sequences enumerating clutters, antichains, hypertrees, and hyperforests, organized by labeling, spanning, and allowance of singletons.

Cf. A006126, A261005, A304996, A304997, A304998, A304999, A305000, A305001.
Cf. A000372, A003182, A014466, A293606, A293993, A306505, A320449, A321679.
Cf. A245079.

Binomial transform of A304999.

Inverse binomial transform of A245079. - Aniruddha Biswas, May 11 2024

a(5)-a(8) from Gus Wiseman, May 31 2018

a(9) from Aniruddha Biswas, May 11 2024

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

A304998 Number of unlabeled antichains of finite sets spanning n vertices without singletons.

Original entry on oeis.org

1, 0, 1, 3, 15, 160, 15963, 489980652

Offset: 0

Gus Wiseman, May 23 2018

nonn

			Non-isomorphic representatives of the a(4) = 15 antichains:
  {{1,2,3,4}}
  {{1,2},{3,4}}
  {{1,4},{2,3,4}}
  {{1,3,4},{2,3,4}}
  {{1,2},{1,3,4},{2,3,4}}
  {{1,3},{1,4},{2,3,4}}
  {{1,3},{2,4},{3,4}}
  {{1,4},{2,4},{3,4}}
  {{1,2,4},{1,3,4},{2,3,4}}
  {{1,2},{1,3},{1,4},{2,3,4}}
  {{1,2},{1,3},{2,4},{3,4}}
  {{1,4},{2,3},{2,4},{3,4}}
  {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}
  {{1,3},{1,4},{2,3},{2,4},{3,4}}
  {{1,2},{1,3},{1,4},{2,3},{2,4},{3,4}}

Cf. A006126, A261005, A304996, A304997, A304998, A304999, A305000, A305001.

a(n > 0) = A261005(n) - A261005(n - 1).

A304985 Number of labeled clutters (connected antichains) spanning n vertices with singleton edges allowed.

Original entry on oeis.org

1, 1, 4, 40, 1344, 203136, 495598592, 309065330371840, 14369391920653644779049472

Offset: 0

Gus Wiseman, May 23 2018

nonn

Only the non-singleton edges are required to form an antichain.

			The a(2) = 4 clutters:
{{1,2}}
{{1},{1,2}}
{{2},{1,2}}
{{1},{2},{1,2}}

Gus Wiseman, Sequences enumerating clutters, antichains, hypertrees, and hyperforests, organized by labeling, spanning, and allowance of singletons.

Cf. A048143, A198085, A261006, A304912, A304981, A304982, A304983, A304984, A304985, A304986.

Cf. A000372, A006126, A014466, A304997, A304999, A306505, A317674, A319721.

For n > 1, a(n) = A048143(n) * 2^n.

A304996 Number of unlabeled antichains of finite sets spanning up to n vertices with singleton edges allowed.

Original entry on oeis.org

1, 2, 6, 24, 166, 3266, 826308

Offset: 0

Gus Wiseman, May 23 2018

			Non-isomorphic representatives of the a(3) = 24 antichains:
{}
{{1}}
{{1,2}}
{{1,2,3}}
{{1},{2}}
{{2},{1,2}}
{{3},{1,2}}
{{3},{1,2,3}}
{{1,3},{2,3}}
{{1},{2},{3}}
{{1},{2},{1,2}}
{{2},{3},{1,3}}
{{2},{3},{1,2,3}}
{{3},{1,2},{2,3}}
{{3},{1,3},{2,3}}
{{1,2},{1,3},{2,3}}
{{1},{2},{3},{2,3}}
{{1},{2},{3},{1,2,3}}
{{2},{3},{1,2},{1,3}}
{{2},{3},{1,3},{2,3}}
{{3},{1,2},{1,3},{2,3}}
{{1},{2},{3},{1,3},{2,3}}
{{2},{3},{1,2},{1,3},{2,3}}
{{1},{2},{3},{1,2},{1,3},{2,3}}

Partial sums of A304997.
Cf. A006126, A261005, A304997, A304998, A304999, A305000, A305001.
Cf. A000372, A014466, A293993, A319639, A319721, A326704, A326972.

a(5)-a(6) from Andrew Howroyd, Aug 14 2019

A304983 Number of unlabeled clutters (connected antichains) spanning n vertices with singleton edges allowed.

Original entry on oeis.org

1, 1, 3, 14, 118, 2916, 819473

Offset: 0

Gus Wiseman, May 23 2018

			Non-isomorphic representatives of the a(3) = 14 clutters:
  {{1,2,3}}
  {{1,3},{2,3}}
  {{3},{1,2,3}}
  {{1,2},{1,3},{2,3}}
  {{3},{1,2},{2,3}}
  {{3},{1,3},{2,3}}
  {{2},{3},{1,2,3}}
  {{1},{2},{3},{1,2,3}}
  {{2},{3},{1,2},{1,3}}
  {{3},{1,2},{1,3},{2,3}}
  {{2},{3},{1,3},{2,3}}
  {{1},{2},{3},{1,3},{2,3}}
  {{2},{3},{1,2},{1,3},{2,3}}
  {{1},{2},{3},{1,2},{1,3},{2,3}}

Cf. A048143, A198085, A261006, A304912, A304981, A304982, A304983, A304984, A304985, A304986, A304997.

Inverse Euler transform of A304997. - Andrew Howroyd, Aug 14 2019

a(5)-a(6) from Andrew Howroyd, Aug 14 2019

A304999 Number of labeled antichains of finite sets spanning n vertices with singleton edges allowed.

Original entry on oeis.org

1, 1, 5, 53, 1577, 212137, 496946349, 309068823607069, 14369391923126237496803793, 146629927766168786109802623629262590838145873

Offset: 0

Gus Wiseman, May 23 2018

Only the non-singleton edges are required to form an antichain.

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

Aniruddha Biswas and Palash Sarkar, Counting unate and balanced monotone Boolean functions, arXiv:2304.14069 [math.CO], 2023.
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. 4.
Gus Wiseman, Sequences enumerating clutters, antichains, hypertrees, and hyperforests, organized by labeling, spanning, and allowance of singletons.

Cf. A006126, A014466, A261005, A304996, A304997, A304998, A304999, A305000, A305001.

Cf. A000372, A003182, A304985, A305844, A306505, A319721, A321679, A324167.

Exponential transform of A304985.

Inverse binomial transform of A305000. - Aniruddha Biswas, May 12 2024

a(5)-a(8) from Gus Wiseman, May 31 2018

a(9) from Aniruddha Biswas, May 12 2024

A318131 Number of non-isomorphic sets of finite (possibly empty) sets with union {1,2,...,n} and intersection {}.

Original entry on oeis.org

1, 1, 6, 60, 3836, 37325360, 25626412263611792, 67516342973185974276922865448446208, 2871827610052485009904013737758920847534777143951264797898686184985092096

Offset: 0

Gus Wiseman, Aug 18 2018

nonn

			Non-isomorphic representatives of the a(2) = 6 sets of sets:
  {{1},{2}}
  {{},{1,2}}
  {{},{1},{2}}
  {{},{1},{1,2}}
  {{1},{2},{1,2}}
  {{},{1},{2},{1,2}}

Andrew Howroyd, Table of n, a(n) for n = 0..12

Cf. A000371, A000612, A003465, A055621, A119563, A131288, A283877, A293606, A304997.

Cf. A318128, A318129, A318130, A318132.

sysnorm[m_]:=If[Union@@m!=Range[Max@@Flatten[m]],sysnorm[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[Sort[sysnorm[m,1]]]];sysnorm[m_,aft_]:=If[Length[Union@@m]<=aft,{m},With[{mx=Table[Count[m,i,{2}],{i,Select[Union@@m,#>=aft&]}]},Union@@(sysnorm[#,aft+1]&/@Union[Table[Map[Sort,m/.{par+aft-1->aft,aft->par+aft-1},{0,1}],{par,First/@Position[mx,Max[mx]]}]])]];
Table[Length[Union[sysnorm/@Select[Subsets[Subsets[Range[n]]],And[Union@@#===Range[n],Intersection@@#=={}]&]]],{n,4}]

a(n) = 2*(A055621(n) - A055621(n-1)) = 2*(A000612(n) - 2*A000612(n-1) + A000612(n-2)) for n >= 2. - Andrew Howroyd, Jan 29 2024

a(5) onwards from Andrew Howroyd, Jan 29 2024

cp's OEIS Frontend

A261005 Number of unlabeled simplicial complexes with n nodes.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Formula

Extensions

A305000 Number of labeled antichains of finite sets spanning some subset of {1,...,n} with singleton edges allowed.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A304998 Number of unlabeled antichains of finite sets spanning n vertices without singletons.

Views

Author

Keywords

Examples

Crossrefs

Formula

A304985 Number of labeled clutters (connected antichains) spanning n vertices with singleton edges allowed.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A304996 Number of unlabeled antichains of finite sets spanning up to n vertices with singleton edges allowed.

Views

Author

Keywords

Examples

Crossrefs

Extensions

A304983 Number of unlabeled clutters (connected antichains) spanning n vertices with singleton edges allowed.

Views

Author

Keywords

Examples

Crossrefs

Formula

Extensions

A304999 Number of labeled antichains of finite sets spanning n vertices with singleton edges allowed.

Views

Author

Keywords

Comments

Examples