Question

question about eulers generating function for partitions

Answer 1

Why exactly does this function expand to this? I'm not familiar with these symbols although I do know how to use it to get partitions.

such as , if we were trying to get the partitions of the integer 4:
We would expand out (1+x+x^2+x^3+x^4)(1+x^2+x^4)(1+x^3)(1+x^4)

the answer would be the co-efficient of the exponent of 4

Answer 2

so while i do understand the pattern of the function, i'm not sure WHY exactly it is so

Answer 3

@Kainui

Answer 4

that is as simplfied as it gets

Answer 5

i thi nkits the sum of hte multiples or something?

Answer 6

That's weird I don't know why the expansion shows the exponents as subscripts, that's not right.

The reason it works is all about how exponents combine. This counts the multiplicity and each of the terms in the product is really a geometric series with different weights.

This is really hard to just "tell" someone. You must actually go and do it and figure this out for yourself.

The most I can tell you is the geometric series has every single number in it, and when you put a specific value of k, that k tells you how much to weigh that number in the partition.