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

2, 1, 2, 5, 20, 180, 16143, 489996795, 1392195548399980210, 789204635842035039135545297410259322

Offset: 0

Author

Colin Mallows

Keywords

• nonn
• nice
• hard

Comments

Also number of pure (= irreducible) group-testing histories of n items - A. Boneh, Mar 31 2000

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

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

The non-spanning/covering case is A003182. The labeled case is A006126. - Gus Wiseman, Feb 20 2019

Examples

			From _Gus Wiseman_, Feb 20 2019: (Start)
Non-isomorphic representatives of the a(0) = 2 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)
		

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 10 2020]
• V. Jovovic and G. Kilibarda, On enumeration of the class of all monotone Boolean functions, in preparation.
• A. A. Mcintosh, personal communication.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• 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. 14.
• V. Jovovic and G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, Diskretnaya Matematika, 11(4) (1999), 127-138 (translated in Discrete Mathematics and Applications, 9(6) (1999), 593-605).
• C. Lienkaemper, When do neural codes come from convex or good covers?, 2015.
• C. L. Mallows, Emails to N. J. A. Sloane, Jun-Jul 1991
• Gus Wiseman, Sequences enumerating clutters, antichains, hypertrees, and hyperforests, organized by labeling, spanning, and allowance of singletons.

Crossrefs

Cf. A000372, A003182, A006126 (labeled case), A007411, A014466, A261005, A293993, A304997, A304998, A304999, A305001, A305855, A306505, A320449, A321679.

Formula

a(n) = A007411(n) + 1.

First differences of A003182. - Gus Wiseman, Feb 23 2019

Extensions

a(6) from A. Boneh, 32 Hantkeh St., Haifa 34608, Israel, Mar 31 2000

Entry revised by N. J. A. Sloane, Jul 23 2006

a(7) from A007411 and A003182. - N. J. A. Sloane, Aug 13 2015

Named edited by Petros Hadjicostas, Apr 08 2020

a(8) from A003182. - Bartlomiej Pawelski, Nov 27 2022

a(9) from A007411. - Dmitry I. Ignatov, Nov 27 2023