Partitions, their classes, and multicolour evenness

Document Type

Article

Publication Date

1-1-2022

Abstract

We extend the theory of maximally even sets to determine the evenness of partitions of the chromatic universe (Formula presented.). Interactions measure the average evenness of colour sets (partitioning sets) of (Formula presented.). For 2-colour partitions the Clough-Douthett maximal-evenness algorithm determines maximally even partitions. But to measure the evenness of non-maximally even partitions, it is necessary to use computational methods. Moreover, for more than two colour sets there is no simple algorithm that determines maximally even partitions. Again, we rely on computational methods. We also explore collections of partitions and partition-classes (orbits under a dihedral group) and construct tables that order partition-classes according to the evenness of their partitions. We use Bell numbers, Stirling numbers of the second kind, and integer partitions to enumerate relevant combinatorial objects related to our investigation.

Publication Source (Journal or Book title)

Journal of Mathematics and Music

First Page

303

Last Page

345

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