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In [[number theory]], the '''partition function''' p(''n'') represents the [[number]] of possible [[integer partition|partitions]] 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. For example, 4 can be partitioned in 5 distinct ways
:4, 3 + 1, 2 + 2, 2 + 1 + 1, 1 + 1 + 1 + 1
So p(4) = 5. By convention p(0) = 1, p(''n'') = 0 for ''n'' negative. Partitions can be graphically visualized with [[Young diagram]]s. They occur in a number of branches of [[mathematics]] and [[physics]], including the study of [[symmetric polynomial]]s, the [[symmetric group]] and in [[group representation|group representation theory]] in general.
==Intermediate function==
One way of getting a handle on the partition function 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:
#smallest [[addend]] is ''k''
#smallest addend is strictly greater than ''k''
The number of partitions meeting the first condition is p(''k'', ''n'' − ''k''). To see this, 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'') since a partition into parts of at least ''k'' which contains no parts of exactly ''k'' must have all parts at least ''k'' + 1.
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==Rademacher's series==
An [[asymptotic]] expression for ''p''(''n'') is given by
:<math>p(n) \sim \frac {\exp \left( \pi \sqrt {2n/3}\right) } {4n\sqrt{3}} \mbox { as } n\rightarrow \infty</math>
This expression was first obtained by [[G. H. Hardy]] and [[Ramanujan]] in [[1918]] and independently by [[J. V. Uspensky]] in [[1920]].
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\pi i s(m,k) - 2\pi inm/k \right)</math>.
Here, the notation <math>(m,n)=1</math> implies that the sum should occur only over the values of ''m'' that are [[relatively prime]] to ''n''. The function <math>s(m,k)</math> is a [[Dedekind sum]]. The proof of Rademacher's formula is interesting in that it involves [[Ford circle]]s, [[Farey sequence]]s, [[modular group|modular symmetry]] and the [[Dedekind eta function]] in a central way.
==Congruences==
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