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''The partition function described here is part of [[number theory]]. The present author has absolutely 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]].''
'''Partition function''' may refer to:
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:
* [[Partition function (statistical mechanics)]], a function used to derive thermodynamic properties
** [[Rotational partition function]], partition function for the rotational modes of a molecule
** [[Vibrational partition function]], partition function for the vibrational modes of a molecule
** [[Partition function (quantum field theory)]], partition function for quantum path integrals
* [[Partition function (mathematics)]], generalization of the statistical mechanics concept
* [[Partition function (number theory)]], the number of possible partitions of an integer
{{disambig}}
1. smallest [[addend]] is k
2. smallest addend is [[strictly greater than]] than k
The number of partitions meeting the first condition is p(k,n-k). If the reason for this is not immediately apparent, imagine a list of all the partitions of the number n-k into numbers of size at least k, then imagine appending "+k" to each partition in the list. Now what is it a list of?
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 cases of this [[recursion|recursive]] function are as follows:
* p(k,n)=0 if k>n
* p(k,n)=1 if k=n
This function will mess with one's [[mind]] if one lets it. Consider the following:
p(1,4)=5
p(2,8)=7
p(3,12)=9
p(4,16)=11
p(5,20)=13
p(6,24)='''16'''
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