# Partition Generating Function

a) Let $$P(x)=\sum_{n=0}^{\infty} p_nx^n=1+x+2x^2+3x^3+5x^4+7x^5+11x^6+\cdots$$ be the partition generating function, and let $Q(x)=\sum_{n=0}^{\infty} q_nx^n$, where $q_n$ is the number of partitions of $n$ containing no $1$s.

Then $\displaystyle\frac{Q(x)}{P(x)}$ is a polynomial. What polynomial is it?

b) Let $P(x)$ be the partition generating function, and let $R(x)=\sum_{n=0}^{\infty} r_nx^n$, where $r_n$ is the number of partitions of $n$ containing no $1$s or $2$s.

Then $\displaystyle \frac{R(x)}{P(x)}$ is a polynomial. What polynomial is it? (Put answer in expanded form)

How can I start this problem?

## 1 Answer

Hint: Try to understand why $$P(x)=\frac{1}{\prod_{n=1}^{\infty}\left(1-x^n\right)},$$ and what are the corresponding expressions for $Q(x)$ and $R(x)$.

Spoiler:

\begin{align*} \frac{1}{1-x}\cdot \frac{1}{1-x^2} \cdot \frac{1}{1-x^3}\cdot\ldots =\,&\Bigl(1+x^1+x^{1+1}+x^{1+1+1}+\ldots\Bigr)\times\\ \times\,&\Bigl(1+x^2+x^{2+2}+x^{2+2+2}+\ldots\Bigr)\times \\ \times\,&\Bigl(1+x^3+x^{3+3}+\ldots\Bigr)\times \ldots=\end{align*} \begin{align*} =1+x^1+\left(x^{1+1}+x^2\right)+\left(x^{1+1+1}+x^{2+1}+x^3\right)+\\+\left(x^{1+1+1+1}+x^{2+1+1}+x^{2+2}+x^{3+1}+x^4\right)+\ldots\end{align*}