This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A006126 M1954 #140 May 02 2025 01:27:52
%S A006126 2,1,2,9,114,6894,7785062,2414627396434,56130437209370320359966,
%T A006126 286386577668298410623295216696338374471993
%N 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.
%C A006126 An antichain cover is a cover such that no element of the cover is a subset of another element of the cover.
%C A006126 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
%C A006126 Also, number of simplicial complexes on an n-element vertex set. - _Richard Stanley_, Feb 10 2019
%C A006126 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
%C A006126 From _Petros Hadjicostas_, Apr 10 2020: (Start)
%C A006126 Hierarchical models are always nonempty because they always include an intercept (or overall effect).
%C A006126 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).
%C A006126 Hierarchical log-linear models for analyzing contingency tables are defined in the classic book by Bishop, Fienberg, and Holland (1975). (End)
%D A006126 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]
%D A006126 V. Jovovic and G. Kilibarda, On enumeration of the class of all monotone Boolean functions, in preparation.
%D A006126 C. L. Mallows, personal communication.
%D A006126 A. A. Mcintosh, personal communication.
%D A006126 R. A. Obando, On the number of nondegenerate monotone boolean functions of n variables, In Preparation.
%D A006126 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006126 R. Baumann and H. Strass, <a href="http://citeseerx.ist.psu.edu/pdf/4c62d4a21e727a9fd40e2def8632df1060342e3a">On the number of bipolar Boolean functions</a>, 2014, preprint.
%H A006126 R. Baumann and H. Strass, <a href="https://doi.org/10.1093/logcom/exx025">On the number of bipolar Boolean functions</a>, Journal of Logic and Computation, 27(8) (2017), 2431-2449.
%H A006126 Aniruddha Biswas and Palash Sarkar, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL28/Biswas/biswas6.html">Counting Unate and Monotone Boolean Functions Under Restrictions of Balancedness and Non-Degeneracy</a>, J. Int. Seq. (2025) Vol. 28, Art. No. 25.3.4. See p. 2.
%H A006126 Florian Bridoux, Amélia Durbec, Kévin Perrot, and Adrien Richard, <a href="https://arxiv.org/abs/2012.02513">Complexity of fixed point counting problems in Boolean Networks</a>, arXiv:2012.02513 [math.CO], 2020.
%H A006126 Florian Bridoux, Nicolas Durbec, Kevin Perrot, and Adrien Richard, <a href="https://doi.org/10.1007/978-3-030-22996-2_12">Complexity of Maximum Fixed Point Problem in Boolean Networks</a>, 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.
%H A006126 K. S. Brown, <a href="http://www.mathpages.com/home/kmath515.htm">Dedekind's problem</a>
%H A006126 Patrick De Causmaecker and Stefan De Wannemacker, <a href="https://arxiv.org/abs/1407.4288">On the number of antichains of sets in a finite universe</a>, arXiv:1407.4288 [math.CO], 2014.
%H A006126 V. Jovovic and G. Kilibarda, <a href="http://mi.mathnet.ru/eng/dm398">On the number of Boolean functions in the Post classes F^{mu}_8</a>, Diskretnaya Matematika, 11 (1999), no. 4, 127-138 (translated in Discrete Mathematics and Applications, 9, (1999), no. 6).
%H A006126 C. L. Mallows, <a href="/A000372/a000372_5.pdf">Emails to N. J. A. Sloane, Jun-Jul 1991</a>
%H A006126 C. L. Mallows & N. J. A. Sloane, <a href="/A006123/a006123.pdf">Emails, May 1991</a>
%H A006126 C. L. Mallows & N. J. A. Sloane, <a href="/A006123/a006123_1.pdf">Emails, Jun. 1991</a>
%H A006126 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Antichain.html">Antichain</a>.
%H A006126 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Cover.html">Cover</a>.
%H A006126 R. I. P. Wickramasinghe, <a href="https://ttu-ir.tdl.org/handle/2346/20089">Topics in log-linear models</a>, 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]
%H A006126 D. H. Wiedemann, <a href="/A000372/a000372.pdf">Letter to N. J. A. Sloane, Nov 03, 1990</a>
%H A006126 D. H. Wiedermann, <a href="/A006126/a006126.pdf">Email to N. J. A. Sloane, May 28 1991</a>
%H A006126 Gus Wiseman, <a href="/A048143/a048143_4.txt">Sequences enumerating clutters, antichains, hypertrees, and hyperforests, organized by labeling, spanning, and allowance of singletons</a>.
%F A006126 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.
%F A006126 Inverse binomial transform of A000372. - _Gus Wiseman_, Feb 24 2019
%e A006126 a(5) = 1 + 90 + 790 + 1895 + 2116 + 1375 + 490 + 115 + 20 + 2 = 6894.
%e A006126 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}}.
%e A006126 From _Gus Wiseman_, Feb 23 2019: (Start)
%e A006126 The a(0) = 2 through a(3) = 9 antichains:
%e A006126 {} {{1}} {{12}} {{123}}
%e A006126 {{}} {{1}{2}} {{1}{23}}
%e A006126 {{2}{13}}
%e A006126 {{3}{12}}
%e A006126 {{12}{13}}
%e A006126 {{12}{23}}
%e A006126 {{13}{23}}
%e A006126 {{1}{2}{3}}
%e A006126 {{12}{13}{23}}
%e A006126 (End)
%t A006126 nn=4;
%t A006126 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]]]];
%t A006126 Table[Length[Select[stableSets[Subsets[Range[n]],SubsetQ],Union@@#==Range[n]&]],{n,0,nn}] (* _Gus Wiseman_, Feb 23 2019 *)
%t A006126 A000372 = Cases[Import["https://oeis.org/A000372/b000372.txt", "Table"], {_, _}][[All, 2]];
%t A006126 lg = Length[A000372];
%t A006126 a372[n_] := If[0 <= n <= lg-1, A000372[[n+1]], 0];
%t A006126 a[n_] := Sum[(-1)^(n-k+1) Binomial[n, k-1] a372[k-1], {k, 0, lg}];
%t A006126 a /@ Range[0, lg-1] (* _Jean-François Alcover_, Jan 07 2020 *)
%Y A006126 Cf. A000372, A056046-A056049, A056052, A056101, A056104, A051112-A051118.
%Y A006126 Cf. A006602, A014466, A261005, A293606, A293993, A305000, A305844, A306550, A307249, A317674, A319721, A320449.
%K A006126 nonn,nice,hard,more
%O A006126 0,1
%A A006126 _N. J. A. Sloane_
%E A006126 Last 3 terms from Michael Bulmer (mrb(AT)maths.uq.edu.au)
%E A006126 Antichain interpretation from _Vladeta Jovovic_ and Goran Kilibarda, Jul 31 2000
%E A006126 a(0) = 2 added by _Gus Wiseman_, Feb 23 2019
%E A006126 Name edited by _Petros Hadjicostas_, Apr 08 2020
%E A006126 a(9) using A000372 added by _Bruno L. O. Andreotti_, May 14 2023