Proving Identities using Partition and Generating Function

I have a problem with these two questions:

1. Let $P_E(n)$ be the number of partitions of $n$ with an even number of parts, $P_O(n)$ the number of partitions of n with an odd number of parts, and $P_D(n)$ the number of partitions of $n$ with distinct odd parts. Show combinatorially that $P_E(n)- P_O(n) = (-1)^n P_D(n)$
2. Show the following equalities.

$\Pi_{i \ge 1}(1-xq^i)^{-1} = \sum_{k \ge 0} {x^k q^k}/{(1-q)(1-q^2)...(1-q^k)}$

$\Pi_{i \ge 1}(1+xq^i) = \sum_{k \ge 0} {x^k q^\binom{k+1}{2}}/{(1-q)(1-q^2)...(1-q^k)}$

What I know is:

Penthagonal Number Theorem(proved it by a bijective proof)

Number of partitions with odd parts, distinct parts

Generating function for $P(n)$ and for the partitions with distinct parts

I don't know much about generating functions and partitions, but I happened to take algebraic combinatorics. Those look quite easy but I couldn't go any further since my basic knowledge is rather shallow.

For #2, I tried to use the identity:

$\sum_{k \ge 0} {1}/{(1-q)(1-q^2)...(1-q^k)} = \sum_{n \ge 0}P_{\le k}(n)q^n$

I just plugged it in the equality and changed the order of the summation, but the result was something not I've wanted. For the second part of the generating function, I thought about using some $q$-binomial coefficient but was rather tough for me now.

I think #1 maybe I can do it only by having some hint about the required bijection.

Thanks!

• The answers to this question give both a combinatorial proof of (1) and one using generating functions. The combinatorial proof is stated pretty concisely, so you’ll probably have to think about it a bit. – Brian M. Scott Nov 4 '14 at 3:07
• Thanks. Both of the proofs were very helpful. – Taxxi Nov 4 '14 at 12:55
• A little late to the party, but I'm having a lot of trouble understanding (2). I know that: $\prod_{i \geq 1}\frac{1}{1-qz^i} = \sum_{k \geq 0} \sum_{n \geq 0} p(n; k)z^n q^k$ where $p(n; k)$ is the number of integer partitions of $n$ into exactly $k$ parts. I'm not sure how to show the first (or second) equations in (2), though. Any help? Should I create my own question for this? – Tyler Durden Oct 25 '16 at 2:31