

A006126


Number of hierarchical models on n labeled factors or variables with linear terms forced. Also number of antichain covers of a labeled nset.
(Formerly M1954)


143




OFFSET

0,1


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 nvariable boolean algebra.  Rodrigo A. Obando (R.Obando(AT)computer.org), Jul 26 2004
Also, number of simplicial complexes on an nelement 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 noncovering 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 loglinear hierarchical models on n labeled factors (categorical variables) with no forcing of terms is given by A000372(n)  1 (Dedekind numbers minus 1).
Hierarchical loglinear models for analyzing contingency tables are defined in the classic book by Bishop, Fienberg, and Holland (1975). (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 loglinear models is defined. It essentially says that if a higherorder parameter term is included in the loglinear model, then all the lowerorder 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

Table of n, a(n) for n=0..8.
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), 24312449.
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, 132143.
K. S. Brown, Dedekind's problem
Patrick De Causmaecker, 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, 127138 (translated in Discrete Mathematics and Applications, 9, (1999), no. 6).
C. L. Mallows, Emails to N. J. A. Sloane, JunJul 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 loglinear models, Master of Science thesis in Statistics, Texas Tech University, Lubbock, TX, 2008. [From the A000372(2)  1 = 4 hierarchical loglinear 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 loglinear models on three factors X, Y, and Z, on p. 36 of his thesis, only Models 1119 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.


FORMULA

a(n) = Sum_{k = 1..C(n, floor(n/2))} b(k, n), where b(k, n) is the number of kantichain covers of a labeled nset.
Inverse binomial transform of A000372.  Gus Wiseman, Feb 24 2019


EXAMPLE

a(5) = 1 + 90 + 790 + 1895 + 2116 + 1375 + 490 + 115 + 20 + 2 = 6894.
There are 9 antichain covers of a labeled 3set: {{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)


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===wQ[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 <= lg1, A000372[[n+1]], 0];
a[n_] := Sum[(1)^(nk+1) Binomial[n, k1] a372[k1], {k, 0, lg}];
a /@ Range[0, lg1] (* JeanFrançois Alcover, Jan 07 2020 *)


CROSSREFS

Cf. A000372, A056046A056049, A056052, A056101, A056104, A051112A051118.
Cf. A006602, A014466, A261005, A293606, A293993, A305000, A305844, A306550, A307249, A317674, A319721, A320449.
Sequence in context: A271574 A274198 A002079 * A078357 A225432 A086382
Adjacent sequences: A006123 A006124 A006125 * A006127 A006128 A006129


KEYWORD

nonn,nice,hard,more


AUTHOR

N. J. A. Sloane


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


STATUS

approved



