A question on partitions of n Let $P$ be the set of partitions of $n$. Let $\lambda$ denote the shape of a particular partition. Let $f_\lambda(i)$ be the frequency of $i$ in $\lambda$ and let $a_\lambda(i) := \# \lbrace j : f_\lambda(j) \geq i \rbrace$. 
For example: $n=5,~ \lambda=(1,1,3),~ f_\lambda(1)=2,~ a_\lambda(1)=2$ (added: since $f_\lambda(1)$ and $f_\lambda(3)$ are both at least 1). It is easy to see that for a fixed $\lambda$, $\sum_k f_\lambda(k)=\sum_k a_\lambda(k)$. But, I am having trouble showing:

For a fixed $k$, $$\sum_\lambda f_\lambda(k)=\sum_\lambda a_\lambda(k)$$

Thanks for the help! 

You can establish a bijection in the partitions of $n$ between parts equal to $k$ and equal parts occurring at least $k$ times by ‘rotating rectangles’ in the diagrams of the partitions.
If $f_\lambda(i)\geqslant k$, we define $\rho_i(\lambda)$ to be the partition resulting from removing $k$ parts equal to $i$ and replacing them with $i$ parts equal to $k$ (i.e. ‘rotating’ an $i\times k$ rectangle in the diagram of $\lambda$, as shown below for $\rho_5$ with $k=3$). We could write $\rho_i(\lambda)=\lambda\:-<\!\!i^k\!\!>+<\!\!k^i\!\!>$.
$\qquad\qquad\qquad\qquad\qquad\qquad$ 
$\rho_i(\lambda)$ is a partition that contains at least $i$ elements equal to $k$. Indeed, if $\mu$ is a partition such that $f_\mu(k)=j$ then there are $j$ distinct partitions, $\lambda_1,\dots,\lambda_j$, with $\rho_i(\lambda_i)=\mu$.
