cp's OEIS Frontend

NP-equivalence classes of unate Boolean functions of n or fewer variables.

Also the number of simple games with n players in minimal winning form up to isomorphism. - Fabián Riquelme, Mar 13 2018

The labeled case is A000372. - Gus Wiseman, Feb 23 2019

First differs from A348260(n + 1) at a(5) = 210, A348260(6) = 233. - Gus Wiseman, Nov 28 2021

Pawelski & Szepietowski show that a(n) = A001206(n) (mod 2) and that a(9) = 6 (mod 210). - Charles R Greathouse IV, Feb 16 2023

The dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}.

The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

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)

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

An intersecting set-system is a finite set of finite nonempty sets (edges), any two of which have a nonempty intersection. The weight of a set-system is the sum of cardinalities of its elements. Weight is generally not the same as number of vertices.

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every set-system (finite set of finite nonempty sets of positive integers) has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18. Elements of a set-system are sometimes called edges.

A set-system is intersecting if no two edges are disjoint. It is an antichain if no edge is a proper subset of any other.

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every set-system (finite set of finite nonempty sets of positive integers) has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18. Elements of a set-system are sometimes called edges. Elements of a set-system are sometimes called edges.

A set-system is intersecting if no two edges are disjoint. It is an antichain if no edge is a proper subset of any other.

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