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| − | A '''partition''' of a [[positive integer]] is a way of expressing it as the sum of other positive integers. Typically, one disregards the order of the summands. For example, there are three partitions of 3: <math>3 = 2+1 =1+1+1</math>.
| + | #REDIRECT[[Partition (disambiguation)]] |
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| − | There is no known, simple formula that gives the number of partitions of a number. There is, however, a rather ugly formula discovered by [[G. H. Hardy]], [[J. E. Littlewood]], and [[Srinivasa Ramanujan]]. However, this formula is rather unwieldy: it is not even known for which values of <math>\displaystyle{n}</math> the number of partitions of <math>\displaystyle{n}</math> is even, despite the presence of a formula!
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| − | A more fruitful way of studying partition numbers is through [[generating function]]s. The generating function for the partitions is given by <math>P(x)=\prod_{n=1}^\infty \frac{1}{1-x^n}</math>. Partitions can also be studied by using the [[Jacobi theta function]], in particular the [[triple product]]. The generating function approach and the theta function approach can be used to study many variants of the partition function, such as the number of ways to write a number ''n'' as the sum of odd parts, or of distinct parts, or of parts congruent to <math> 1\pmod 3</math>, etc.
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| − | == Resources ==
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| − | * [http://www.artofproblemsolving.com/Resources/Papers/LaurendiPartitions.pdf Partitions of Integers by Joseph Laurendi]
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| − | * [http://www.albanyconsort.com/JacobiTheta/JacobiTheta.pdf The Jacobi Theta Function by Simon Rubinstein-Salzedo]
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