TITLE: Same number of partitions of a certain type?
QUESTION [5 upvotes]: Is there a quick explanation of why the number of partitions of $n$ such that no parts are divisible by $d$ is the same as the number of partitions of $n$ where no part is repeated $d$ or more times, for all $n$ and $d$?
I'm having difficulty jumping from the conditions of one to the other to see why they should give the same number of partitions. Cheers.

REPLY [7 votes]: A bijective argument:
On the side without too many repetitions, you break up parts divisible by $d^k$ into $d^k$ parts (where $d^k$ is the highest power of $d$ dividing the size of the part). This gets you a partition without parts divisible by $d$.
To go back, you write the multiplicities in base $d$ and if you have $\sum_i a_i d^i$ parts of size $s$, then you glue them, to get $a_i$ parts of size $sd^i$.