Partition function: Difference between revisions

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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]].''

The partition [[function]] p(''n'') is a [[multiplicative function|non-multiplicative function]] and 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:

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'''