cp's OEIS Frontend

From Gus Wiseman, Jul 03 2019: (Start)

Also the number of antichains covering n vertices and having empty intersection (meaning there is no vertex in common to all the edges). For example, the a(3) = 5 antichains are:

{{3},{1,2}}

{{2},{1,3}}

{{1},{2,3}}

{{1},{2},{3}}

{{1,2},{1,3},{2,3}}

(End)

No edge of a minimal cover can be a subset of any other, so minimal covers are antichains, but the converse is not true. - Gus Wiseman, Jul 03 2019

a(n) is the number of undirected graphs on n nodes for which the intersection number and independence number are equal. See Proposition 2.3.7 and Theorem 2.3.3 of the Deligeorgaki et al. paper below. - Alex Markham, Oct 13 2022

A set system (set of sets) is an antichain if no element is a subset of any other.

From Gus Wiseman, Jul 02 2019: (Start)

If self-dual means (pairwise) intersecting, then a(n) is the number of maximal intersecting antichains of nonempty subsets of {1..(n - 1)}. A set of sets is an antichain if no part is a subset of any other, and is intersecting if no two parts are disjoint. For example, the a(2) = 1 through a(5) = 20 maximal intersecting antichains are:

{1} {1} {1} {1}

{2} {2} {2}

{12} {3} {3}

{123} {4}

{12}{13}{23} {1234}

{12}{13}{23}

{12}{14}{24}

{13}{14}{34}

{23}{24}{34}

{12}{134}{234}

{13}{124}{234}

{14}{123}{234}

{23}{124}{134}

{24}{123}{134}

{34}{123}{124}

{12}{13}{14}{234}

{12}{23}{24}{134}

{13}{23}{34}{124}

{14}{24}{34}{123}

{123}{124}{134}{234}

(End)

Two or more numbers are pairwise coprime if no pair of them has a common divisor > 1.

A set system (set of sets) is an antichain if no element is a subset of any other, and is intersecting if no two element are disjoint.

Covering means there are no isolated vertices. A set system (set of sets) is an antichain if no part is a subset of any other, and is intersecting if no two parts are disjoint.

Also the number of maximal pairwise coprime sets of divisors > 1 of n. For example, the a(n) sets for n = 12, 30, 36, 60, 120 are:

{6} {30} {6} {30} {30}

{12} {2,15} {12} {60} {60}

{2,3} {3,10} {18} {2,15} {120}

{3,4} {5,6} {36} {3,10} {2,15}

{2,3,5} {2,3} {3,20} {3,10}

{2,9} {4,15} {3,20}

{3,4} {5,6} {3,40}

{4,9} {5,12} {4,15}

{2,3,5} {5,6}

{3,4,5} {5,12}

{5,24}

{8,15}

{2,3,5}

{3,4,5}

{3,5,8}

A set system (set of sets) is an antichain if no part is a subset of any other, and is intersecting if no two parts are disjoint.

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