Difference between revisions of "Partition"
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| − | There is no known, simple formula that gives the number of partitions of a number. | + | 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 ''n'' is the number of partitions of ''n'' 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]]. | ||
== Resources == | == Resources == | ||
* [http://www.artofproblemsolving.com/Resources/Papers/LaurendiPartitions.pdf Partitions of Integers by Joseph Laurendi] | * [http://www.artofproblemsolving.com/Resources/Papers/LaurendiPartitions.pdf Partitions of Integers by Joseph Laurendi] | ||
| + | * [http://www.albanyconsort.com/JacobiTheta/JacobiTheta.pdf The Jacobi Theta Function by Simon Rubinstein-Salzedo] | ||
Revision as of 23:56, 19 June 2006
A partition of a number is a way of expressing it as the sum of some number of positive integers. For example, the partitions of 3 are: 3, 2+1, and 1+1+1 (notice how the order of the addends is disregarded).
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 n is the number of partitions of n even, despite the presence of a formula!
A more fruitful way of studying partition numbers is through generating functions. The generating function for the partitions is given by 
. Partitions can also be studied by using the Jacobi theta function, in particular the triple product.
