Let $n,k\in\mathbb{Z}^+$ and $n\geq k$. Suppose $n = \lambda_1 + \lambda_2 + \cdots + \lambda_k$ is an integer partition of $n$, and $\lambda_1 \geq\lambda_2\geq\cdots\geq\lambda_k$.
Show (the number of partitions of $n$ in which $\lambda_1=k$) = (the number of partitions of $n-k$ in which $\mu_i\leq k,\forall i$)
So I think first to derive the generating functions for both sides and then prove they're equal to each other? And the generating functions of $p(n)$ is $P(n)=\sum\limits_{n=0}^\infty p_nx^n=\prod\limits_{i=1}^\infty \frac{1}{1-x^i}$?
There's an obvious bijection $\{\lambda_1 = k,\lambda_2,\ldots,\lambda_k\} \leftrightarrow \{\mu_1,\ldots,\mu_{k-1}\} = \{\lambda_2,\ldots,\lambda_k\}$ between partitions of $n$ into $k$ elements $\lambda_i$, $i=1,\ldots,k$ with $\lambda_1 = k$, $\lambda_1 \geq \ldots \geq \lambda_k$ and partitions of $n-k$ into $k-1$ elements $\mu_j$, $j=1,\ldots,{k-1}$ with all elements less than or equal to $k$ AND $\mu_1 \geq \mu_2 \geq \ldots \mu_{k-1}$. Indeed, $k = \lambda_1 \geq \lambda_i$ for all $\lambda_i$ with $i \geq 2$ and $\lambda_2 + \ldots + \lambda_k = n - k$.