Content deleted Content added
m ortho |
|||
Line 39:
:<i>n</i> = <i>a</i><sub>1</sub> + 2<i>a</i><sub>2</sub> + 3<i>a</i><sub>3</sub> + ... = (1 + 1 + 1 + 1 + ...) + (2 + 2 + 2 + 2 + ...) + (3 + 3 + 3 ...) + ...,
where each number <i>i</i> appears <i>a</i><sub>i</sub> times. This is precisely the definition of a partition of <i>n</i>, so our product is the desired generating function. More generally, the generating function for the partitions of <i>n</i> into numbers from a set <i>A</i> can be found by taking only those terms in the product where <i>k</i> is an element of <i>A</i>. This result is due to [[Leonhard Euler|Euler]]
The formulation of the generating function is similar to the product formulation of many [[modular form|modular forms]], giving some idea of the connection between the two. It can also be used in conjunction with the [[pentagonal number theorem]] to derive a recurrence for the partition function stating that
| |||