Antichains and counterpoint dichotomies

TL;DR

This paper constructs a special antichain using group theory to bound its size and applies this to limit the number of strong counterpoint dichotomies under affine transformations.

Contribution

It introduces a novel group-theoretical method to analyze antichains and applies it to bound the count of strong counterpoint dichotomies.

Findings

Established an upper bound on the size of a special antichain.

Bounded the number of strong counterpoint dichotomies up to affine isomorphisms.

Demonstrated the application of group theory to combinatorial structures.

Abstract

We construct a special type of antichain (i. e., a family of subsets of a set, such that no subset is contained in another) using group-theoretical considerations, and obtain an upper bound on the cardinality of such an antichain. We apply the result to bound the number of strong counterpoint dichotomies up to affine isomorphisms.