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

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• hard
• more

Comments

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)

Examples

			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)
		

References

• 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).

Links

• 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.

Crossrefs

Cf. A000372, A056046-A056049, A056052, A056101, A056104, A051112-A051118.

Cf. A006602, A014466, A261005, A293606, A293993, A305000, A305844, A306550, A307249, A317674, A319721, A320449.

Programs

Mathematica

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 *)

Formula

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

Extensions

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