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%I #36 Nov 27 2023 16:01:29
%S 1,2,4,9,29,209,16352,490013147,1392195548889993357,
%T 789204635842035040527740846300252679
%N Number of non-isomorphic antichains of nonempty subsets of {1,...,n}.
%C The spanning case is A006602 or A261005. The labeled case is A014466.
%C From _Petros Hadjicostas_, Apr 22 2020: (Start)
%C a(n) is the number of "types" of log-linear hierarchical models on n factors in the sense of _Colin Mallows_ (see the emails to _N. J. A. Sloane_).
%C Two hierarchical models on n factors belong to the same "type" iff one can obtained from the other by a permutation of the factors.
%C The total number of hierarchical log-linear models on n factors (in all "types") is given by A014466(n) = A000372(n) - 1.
%C The name of a hierarchical log-linear model on factors is based on the collection of maximal interaction terms, which must be an antichain (by the definition of maximality).
%C In his example on p. 1, _Colin Mallows_ groups the A014466(3) = 19 hierarchical log-linear models on n = 3 factors x, y, z into a(3) = 9 types. See my example below for more details. (End)
%C First differs from A348260(n + 1) - 1 at a(5) = 209, A348260(6) - 1 = 232. - _Gus Wiseman_, Nov 28 2021
%H C. L. Mallows, <a href="/A000372/a000372_5.pdf">Emails to N. J. A. Sloane, Jun-Jul 1991</a>, p. 1.
%H R. I. P. Wickramasinghe, <a href="http://hdl.handle.net/2346/20089">Topics in log-linear models</a>, Master of Science thesis in Statistics, Texas Tech University, Lubbock, TX, 2008, p. 36.
%H 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 a(n) = A003182(n) - 1.
%F Partial sums of A006602 minus 1.
%e Non-isomorphic representatives of the a(0) = 1 through a(3) = 9 antichains:
%e {} {} {} {}
%e {{1}} {{1}} {{1}}
%e {{1,2}} {{1,2}}
%e {{1},{2}} {{1},{2}}
%e {{1,2,3}}
%e {{1},{2,3}}
%e {{1},{2},{3}}
%e {{1,3},{2,3}}
%e {{1,2},{1,3},{2,3}}
%e From _Petros Hadjicostas_, Apr 23 2020: (Start)
%e We expand _Colin Mallows_'s example from p. 1 of his list of 1991 emails. For n = 3, we have the following a(3) = 9 "types" of log-linear hierarchical models:
%e Type 1: ( ), Type 2: (x), (y), (z), Type 3: (x,y), (y,z), (z,x), Type 4: (x,y,z), Type 5: (xy), (yz), (zx), Type 6: (xy,z), (yz,x), (zx,y), Type 7: (xy,xz), (yx,yz), (zx,zy), Type 8: (xy,yz,zx), Type 9: (xyz).
%e For each model, the name only contains the maximal terms. See p. 36 in Wickramasinghe (2008) for the full description of the 19 models.
%e Strictly speaking, I should have used set notation (rather than parentheses) for the name of each model, but I follow the tradition of the theory of log-linear models. In addition, in an interaction term such as xy, the order of the factors is irrelevant.
%e Models in the same type essentially have similar statistical properties.
%e For example, models in Type 7 have the property that two factors are conditionally independent of one another given each level (= category) of the third factor.
%e Models in Type 6 are such that two factors are jointly independent from the third one. (End)
%Y Cf. A000372, A003182, A006126, A006602, A014466, A261005, A293606, A293993, A304996, A305000, A305001, A305857, A317674, A319721, A320449, A321679.
%Y Cf. A007363, A306007, A307249, A326358, A326359, A326360, A326363.
%K nonn,more
%O 0,2
%A _Gus Wiseman_, Feb 20 2019
%E a(8) from A003182. - _Bartlomiej Pawelski_, Nov 27 2022
%E a(9) from A003182. - _Dmitry I. Ignatov_, Nov 27 2023
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