Give the generating function for the number of partitions of an integer into parts of size one through ten

Question

I have no idea where to start. Could I get some help?

Answer 1

The generating function for the unrestricted partition function $p(n)$ is given by $$\sum_{n=0}^{\infty}p(n)q^n = \prod_{j=1}^{\infty}\frac{1}{1 - q^n}.$$ To see this, expand each factor in the infinite product as a geometric series \begin{equation*} \begin{aligned} \prod_{j=1}^{\infty}\frac{1}{1 - q^n} &= \prod_{j=1}^{\infty}(1 + q^{nj} + q^{2nj} + \cdots) \\ &= (1 + q + q^2 + \cdots)(1 + q^2 + q^4 + \cdots)(1 + q^3 + q^6 + \cdots). \end{aligned} \end{equation*}

Any term in the series is given by multiplying together powers of $q$, one chosen from each factor in the last line above, where all but finitely many are chosen to be $1$. The power of $q$ chosen from the first factor represents the number of ones in the corresponding partition, the power chosen from the second factor represents the number of twos, and so on. So if you want to restrict to partitions whose parts have size at most $10$, what should you do?