(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 $\frac{Q(x)}{P(x)}$ 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 $\frac{R(x)}{P(x)}$ is a polynomial. What polynomial is it? (Enter your answer in expanded form.)
I solved part (a) by listing all the partitions and the answer was 1-x. I tried to do the same for part (b), but I'm having trouble. Can anyone help? Thanks
Best Answer
From the definition, it should be clear that \begin{eqnarray} P(x) &=& \prod_{k \geq 1}(1 + x^k + x^{2k} + \dots) = \prod_{k \geq 1} \frac 1{1 - x^k}\\ Q(x) &=& \prod_{k \geq 2}(1 + x^k + x^{2k} + \dots) = \prod_{k \geq 2} \frac 1{1 - x^k}\\ R(x) &=& \prod_{k \geq 3}(1 + x^k + x^{2k} + \dots) = \prod_{k \geq 3} \frac 1{1 - x^k}\\ \end{eqnarray} and it follows easily that $\frac{Q(x)}{P(x)} = 1 - x$ and $\frac{R(x)}{P(x)} = (1 - x)(1 - x^2) = 1 - x - x^2 + x^3$.