Question

Can someone explain how generating function work for partitions?

Answer 1

im not too familiar with series or anything so is there anything i should read before hand

Answer 2

like how does that give me partitions

Answer 3

@SolomonZelman @jim_thompson5910

Answer 4

@satellite73

Answer 5

I suggest you review this doc on generating functions first
http://db.math.ust.hk/notes_download/elementary/algebra/ae_A11.pdf

Answer 6

I am really starting to get this. However, I have a question about removing some terms when finding the partitions. You said for example that that expansion for the integer 4, we can remove (1+x^3), what are the rules for that?

like how would i completely simplify for the integer 9?

(1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9)(1+x^2+x^4+x^6+x^8)
(1+x^3+x^6+x^9)(1+x^4+x^8)(1+x^5)(1+x^6)(1+x^7)(1+x^8)(1+x^9)

http://openstudy.com/users/dtan5457#/updates/566a542ee4b090814d903863

from that thread @rational

Answer 7

Would you agree that we get a "polynomial"` after expanding below product ?

(1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9)(1+x^2+x^4+x^6+x^8)
(1+x^3+x^6+x^9)(1+x^4+x^8)(1+x^5)(1+x^6)(1+x^7)(1+x^8)(1+x^9)

Answer 8

yes and i believe we look for the co-efficient of the term with the exponent of 9

Answer 9

thats right, what is the coefficent of x^9 in the simplified polynomial ?

Answer 10

30

Answer 11

see if there is anything else I should know. To be honest, I only get the pattern of the expansions but Im still not sure what the symbols on the function mean and why exactly they expand the way they do

Answer 12

what symbols do you not understand ?

Answer 13

generally i am unfamiliar with series and sigma symbols ( i need to review them)

i think the things i totally did not learn the is symbol next to the equal sign

Answer 14

\(\prod \) is just a shorthand for product

Answer 15

for example,

\[\prod\limits_{k=1}^3k = 1\cdot 2\cdot 3 = 6\]