Sunday, January 13, 2013

Multiset partitions

Every multiset can be partitioned into a multiset of parts . The mathematical structure of a multisets partition system is determined by the collection of its multiplicities:
{:x 2, :y 2, :z 3}
{2 2, 3 1}
Multiplicity sets of {1 n} describe sets and multiplicity multisets of {n 1} describe additive partitions.

Additive partitions:
Additive partitions are partitions of the multiset {1 n}.
{1 4}, {1 2, 2 1}, {1 3}, {2 2}, {4 1}
Every multiset that contains only a single element has partitions isomorphic to the additive partitions.

Set partitions:
Multisets whose multiplicities are all one can be described as sets:
{:x 1, :y 1, :z 1}
#{#{:x} #{:y :z}}
In Clojure, sets can be described using the pound sign # leaving out the multiplicity values of one.  

Multiplicative partitions:
Every positive integer can be factored into a multiset of prime numbers. The multiplicities multiset of the prime factorization is known as the numbers prime signature.
(= (factors 24) {2 3, 3 1})
Multiplicative partitions can be used to describe association structures in terms of the size of each place in the structure.

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