Difference between revisions of "Partition"
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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>. | 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>. | ||
| − | 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, despite the presence of a formula! | + | 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! |
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. | 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. | ||
Revision as of 16:24, 3 July 2006
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: 
.
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 
the number of partitions of is 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. 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 
, etc.
