A306505 Number of non-isomorphic antichains of nonempty subsets of {1,...,n}.

1, 2, 4, 9, 29, 209, 16352, 490013147, 1392195548889993357, 789204635842035040527740846300252679

Offset

0,2

Comments

The spanning case is A006602 or A261005. The labeled case is A014466.

From Petros Hadjicostas, Apr 22 2020: (Start)

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).

Two hierarchical models on n factors belong to the same "type" iff one can obtained from the other by a permutation of the factors.

The total number of hierarchical log-linear models on n factors (in all "types") is given by A014466(n) = A000372(n) - 1.

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).

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)

First differs from A348260(n + 1) - 1 at a(5) = 209, A348260(6) - 1 = 232. - Gus Wiseman, Nov 28 2021

Links

Table of n, a(n) for n=0..9.

C. L. Mallows, Emails to N. J. A. Sloane, Jun-Jul 1991, p. 1.

R. I. P. Wickramasinghe, Topics in log-linear models, Master of Science thesis in Statistics, Texas Tech University, Lubbock, TX, 2008, p. 36.

Gus Wiseman, Sequences enumerating clutters, antichains, hypertrees, and hyperforests, organized by labeling, spanning, and allowance of singletons.

Formula

a(n) = A003182(n) - 1.

Partial sums of A006602 minus 1.

Example

Non-isomorphic representatives of the a(0) = 1 through a(3) = 9 antichains:
  {}  {}     {}         {}
      {{1}}  {{1}}      {{1}}
             {{1,2}}    {{1,2}}
             {{1},{2}}  {{1},{2}}
                        {{1,2,3}}
                        {{1},{2,3}}
                        {{1},{2},{3}}
                        {{1,3},{2,3}}
                        {{1,2},{1,3},{2,3}}
From Petros Hadjicostas, Apr 23 2020: (Start)
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:
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).
For each model, the name only contains the maximal terms. See p. 36 in Wickramasinghe (2008) for the full description of the 19 models.
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.
Models in the same type essentially have similar statistical properties.
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.
Models in Type 6 are such that two factors are jointly independent from the third one. (End)

Crossrefs

Cf. A000372, A003182, A006126, A006602, A014466, A261005, A293606, A293993, A304996, A305000, A305001, A305857, A317674, A319721, A320449, A321679.

Cf. A007363, A306007, A307249, A326358, A326359, A326360, A326363.

Sequence in context: A300491 A229686 A208965 * A243789 A214935 A092329

Adjacent sequences: A306502 A306503 A306504 * A306506 A306507 A306508

Keyword

nonn,more

Author

Gus Wiseman, Feb 20 2019

Extensions

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

a(9) from A003182. - Dmitry I. Ignatov, Nov 27 2023

Status

approved