How many ways can one divide $n-m$ into parts of at most size $m$?
The only way I've thought of doing this is by summing up all the ways from parts of size $1$ to size $m$. The problem is that I don't know the size of $m$ relative to $n$.
How many ways can one divide $n-m$ into parts of at most size $m$?
The only way I've thought of doing this is by summing up all the ways from parts of size $1$ to size $m$. The problem is that I don't know the size of $m$ relative to $n$.
The number of partitions of $n-m$ into parts of size at most $m$ is the coefficient of $x^{n-m}$ in $\frac{1}{(1-x)(1-x^2)\dots(1-x^m)}.$
Depending on the application, one might prefer a recursive definition which can be derived from the above expression.
For the application mentioned in the other answer, however, it is best to use Ferrers diagrams as mentioned in the comment to the other answer.
The original question is actually from number theory: show that the number of partitions of n into exactly m parts is equal to the number of partitions of n-m into parts of size at most m. So I guess the order does not matter.