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Multiset Partitions

A multiset is an orderless collection of elements in which elements may be repeated, as in

. In this Demonstration, the elements of the multiset are arranged at the corners of a regular

-gon. Elements in the same submultiset are connected with line segments (singletons appear as dots).

(17 lines omitted)

Snapshot 1: when a multiset containing

elements is a set—that is, no elements are repeated—the number of partitions is the

Bell number, calculated with the Mathematica function BellB

Snapshot 2: at the other extreme is a multiset consisting of the same element repeated

times, which has

distinct partitions

, where

is the number of partitions of the integer

, calculated with the Mathematica function PartitionsP

Snapshot 3: the general case has more than two elements with some element occurring more than once

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