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

%I A006602 M1532 #72 May 02 2025 01:27:05
%S A006602 2,1,2,5,20,180,16143,489996795,1392195548399980210,
%T A006602 789204635842035039135545297410259322
%N A006602 a(n) is the number of hierarchical models on n unlabeled factors or variables with linear terms forced.
%C A006602 Also number of pure (= irreducible) group-testing histories of n items - A. Boneh, Mar 31 2000
%C A006602 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
%C A006602 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
%C A006602 The non-spanning/covering case is A003182. The labeled case is A006126. - _Gus Wiseman_, Feb 20 2019
%D A006602 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]
%D A006602 V. Jovovic and G. Kilibarda, On enumeration of the class of all monotone Boolean functions, in preparation.
%D A006602 A. A. Mcintosh, personal communication.
%D A006602 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006602 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. 14.
%H A006602 V. Jovovic and G. Kilibarda, <a href="http://mi.mathnet.ru/eng/dm/v11/i4/p127">On the number of Boolean functions in the Post classes F^{mu}_8</a>, Diskretnaya Matematika, 11(4) (1999), 127-138 (<a href="https://doi.org/10.1515/dma.1999.9.6.593">translated</a> in Discrete Mathematics and Applications, 9(6) (1999), 593-605).
%H A006602 C. Lienkaemper, <a href="http://www.math.tamu.edu/REU/results/REU_2015/lienreport.pdf">When do neural codes come from convex or good covers?</a>, 2015.
%H A006602 C. L. Mallows, <a href="/A000372/a000372_5.pdf">Emails to N. J. A. Sloane, Jun-Jul 1991</a>
%H A006602 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 A006602 a(n) = A007411(n) + 1.
%F A006602 First differences of A003182. - _Gus Wiseman_, Feb 23 2019
%e A006602 From _Gus Wiseman_, Feb 20 2019: (Start)
%e A006602 Non-isomorphic representatives of the a(0) = 2 through a(4) = 20 antichains:
%e A006602   {}    {{1}}  {{12}}    {{123}}         {{1234}}
%e A006602   {{}}         {{1}{2}}  {{1}{23}}       {{1}{234}}
%e A006602                          {{13}{23}}      {{12}{34}}
%e A006602                          {{1}{2}{3}}     {{14}{234}}
%e A006602                          {{12}{13}{23}}  {{1}{2}{34}}
%e A006602                                          {{134}{234}}
%e A006602                                          {{1}{24}{34}}
%e A006602                                          {{1}{2}{3}{4}}
%e A006602                                          {{13}{24}{34}}
%e A006602                                          {{14}{24}{34}}
%e A006602                                          {{13}{14}{234}}
%e A006602                                          {{12}{134}{234}}
%e A006602                                          {{1}{23}{24}{34}}
%e A006602                                          {{124}{134}{234}}
%e A006602                                          {{12}{13}{24}{34}}
%e A006602                                          {{14}{23}{24}{34}}
%e A006602                                          {{12}{13}{14}{234}}
%e A006602                                          {{123}{124}{134}{234}}
%e A006602                                          {{13}{14}{23}{24}{34}}
%e A006602                                          {{12}{13}{14}{23}{24}{34}}
%e A006602 (End)
%Y A006602 Cf. A000372, A003182, A006126 (labeled case), A007411, A014466, A261005, A293993, A304997, A304998, A304999, A305001, A305855, A306505, A320449, A321679.
%K A006602 nonn,nice,hard
%O A006602 0,1
%A A006602 _Colin Mallows_
%E A006602 a(6) from A. Boneh, 32 Hantkeh St., Haifa 34608, Israel, Mar 31 2000
%E A006602 Entry revised by _N. J. A. Sloane_, Jul 23 2006
%E A006602 a(7) from A007411 and A003182. - _N. J. A. Sloane_, Aug 13 2015
%E A006602 Named edited by _Petros Hadjicostas_, Apr 08 2020
%E A006602 a(8) from A003182. - _Bartlomiej Pawelski_, Nov 27 2022
%E A006602 a(9) from A007411. - _Dmitry I. Ignatov_, Nov 27 2023