A306550 -id:A306550 - 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

A261005 Number of unlabeled simplicial complexes with n nodes.

Original entry on oeis.org

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

A307249 Number of simplicial complexes with n nodes.

Original entry on oeis.org

1, 1, 2, 9, 114, 6894, 7785062, 2414627396434, 56130437209370320359966, 286386577668298410623295216696338374471993

Offset: 0

Gus Wiseman, Mar 31 2019

Except for a(0) = 1, this is also the number of antichains of nonempty sets covering n vertices (A006126). There are two antichains of size zero, namely {} and {{}}, while there is only one simplicial complex, namely {}. The unlabeled case is A261005. The non-covering case is A014466.

			Maximal simplices of the a(0) = 1 through a(3) = 9 simplicial complexes:
  {}    {{1}}  {{12}}    {{123}}
               {{1}{2}}  {{1}{23}}
                         {{2}{13}}
                         {{3}{12}}
                         {{12}{13}}
                         {{12}{23}}
                         {{13}{23}}
                         {{1}{2}{3}}
                         {{12}{13}{23}}

Francisco Ponce Carrión and Seth Sullivant, Marginal Independence and Partial Set Partitions, arXiv:2402.16292 [math.ST], 2024. See p. 21.
Gus Wiseman, Sequences enumerating clutters, antichains, hypertrees, and hyperforests, organized by labeling, spanning, and allowance of singletons.

Cf. A000372, A003182, A006126, A006602, A014466, A261005, A293606, A293993, A305000, A305844, A306550, A317674, A319721, A320449.

nn=5;
stableSets[u_,Q_]:=If[Length[u]===0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r===w||Q[r,w]||Q[w,r]],Q]]]];
Table[Length[stableSets[Subsets[Range[n],{2,n}],SubsetQ]],{n,0,nn}]

Inverse binomial transform of A014466.

a(9) from Dmitry I. Ignatov, Nov 25 2023

A322291 Triangle T read by rows: T(n, k) = Sum_{i=1..k} binomial(n, floor((n-k)/2)+i).

Original entry on oeis.org

1, 2, 3, 3, 6, 7, 6, 10, 14, 15, 10, 20, 25, 30, 31, 20, 35, 50, 56, 62, 63, 35, 70, 91, 112, 119, 126, 127, 70, 126, 182, 210, 238, 246, 254, 255, 126, 252, 336, 420, 456, 492, 501, 510, 511, 252, 462, 672, 792, 912, 957, 1002, 1012, 1022, 1023, 462, 924, 1254, 1584, 1749, 1914, 1969, 2024, 2035, 2046, 2047

Offset: 1

Stefano Spezia, Aug 28 2019

T(n, k) is a sharp upper bound on the cardinality of a k-antichain in {0, 1}^n due to P. Erdős.

T(n, k) is also the total number of compositions with first part k, n+1 parts, and all differences between adjacent parts in {-1,1}. - John Tyler Rascoe, May 07 2023

			n\k|   1    2    3    4    5    6
---+-----------------------------
1  |   1
2  |   2    3
3  |   3    6    7
4  |   6   10   14   15
5  |  10   20   25   30   31
6  |  20   35   50   56   62   63
...

John Tyler Rascoe, Rows n = 1..141 of the triangle, flattened
P. Erdős, On a lemma of Littlewood and Offord, Bull. Amer. Math. Soc., 51 (1945), 898-902.
C. Pelekis and V. Vlasák, On k-antichains in the unit n-cube, arXiv:1908.04727 [math.CA], 2019.

Cf. A000225 (diagonal), A189390 (row sums).

Cf. A000247, A000918, A001405, A006126, A007318, A052515, A052516, A263857, A272352, A306550.

Flat(List([1..11], n->List([1..n], k->Sum([1..k], i->Binomial(n, Int((n-k)/2)+i)))));

a:=(n, k)->sum(binomial(n, floor((1/2)*n-(1/2)*k)+i), i = 1..k): seq(seq(a(n, k), k = 1..n), n = 1..11);

T[n_,k_]:=Sum[Binomial[n,Floor[(n-k)/2]+i],{i,1,k}]; Table[T[n,k],{n,1,11},{k,1,n}]

T(n, k) = sum(i=1, k, binomial(n, floor((n-k)/2)+i));

T(n, n) = A000225(n).
T(n, n-1) = A000918(n).
T(n, n-2) = A000247(n).
T(n, n-3) = A052515(n).
T(n, n-4) = A272352(n+1).
T(n, n-5) = A052516(n).

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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A261005 Number of unlabeled simplicial complexes with n nodes.

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A307249 Number of simplicial complexes with n nodes.

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A322291 Triangle T read by rows: T(n, k) = Sum_{i=1..k} binomial(n, floor((n-k)/2)+i).

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