# Building a Generating Function to Represent an Integer Partition

From a Miklos Bona combinatorics textbook,

I'm at an almost total loss. My professor recently discussed products of generating functions, so I suspected this problem might relate. The only strategy I came up with was to find a g.f. for $$p_{e} (n)$$, that is, the number of partitions of $$n$$ into even parts that are at most 6, and a separate one, say $$p_{o} (n)$$, or the number of ways of partitions of $$n$$ into at most one odd part, and multiply them.

For $$p_{e} (n)$$, I obtained $$p_{e} (n)=1+x^{2}+2x^{4}+3x^{6}+...$$ ... which seems fishy. For $$p_{o} (n)$$, I obtained $$p_{o} (n)=x+x^{3}+x^{5}+...$$... which again I'm not sure if that's quite right. I truly want to learn to do these types of problems, so any help just getting started would be much appreciated. Also, sorry for not posting more preliminary work - this thing has me on the ropes.

• Images of text are not very good for search engines. Would you mind typing up the question? You can indicate that it is a quote by preceding the relevant lines with >. – Peter Taylor Oct 11 '18 at 7:20

The GF for partitions into $$2$$s, $$4$$s and $$6$$s is $$f(x)=\frac1{(1-x^2)(1-x^4)(1-x^6)}.$$ The GF for partitions into exactly one odd part is $$g(x)=x+x^3+x^5+\cdots=\frac x{1-x^2}.$$ The GF for partitions into exactly one odd part and other parts in $$\{2,4,6\}$$ is $$f(x)g(x)$$ etc.