Partition function: Difference between revisions

Revision as of 18:24, 3 June 2002

Revision as of 17:37, 23 August 2002

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''The partition function described here is part of [[number theory]]. The present author has absotelyabsolutely no idea whether this is the same function referred to as a partition function in [[derivation of the partition function|statistical mechanics]] or [[partition function game|game theory]].''

The partition [[function]] p(n) represents the [[number]] of possible [[partition]]s of a [[natural number]] n, which is to say the number of distinct (and order independent) ways of representing ''n'' as a [[sum]] of natural numbers. The partition function is easy to calculate. One way of doing so involves an intermediate function p(k,n) which represents the number of partitions of n using only natural numbers at least as large as k. For any given value of k, partitions counted by p(k,n) fit into exactly one of the following categories:

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The number of partitions meeting the second condition is p(k+1,n). Can anyone explain to us why?

Since the two conditions are [[mutually exclusive]], the number of partitions meeting either condition is p(k+1,n)+p(k,n-k). The base casecases of this [[recursion|recursive]] function are as follows:

* p(k,n)=0 if k>n