To elaborate a little on joriki's and A. De Luca's helpful comments, there is a bijection between what you are seeking (partitions of $X$ into $N$ parts) and partitions of $X - N$ into at most $N$ parts. To see this, you can visualize your partition $\lambda_1 \geq \cdots \lambda_N \geq 1$ as a triangular array of boxes, with $\lambda_1$ boxes in the first row, $\lambda_2$ boxes in the second row, etc. (with the whole diagram left-justified). To say that it is a partition of $X$ means that $\lambda_1 + \cdots + \lambda_N = X$, or equivalently that there are $X$ boxes in the diagram. To say that it has exactly $N$ parts means that the diagram has exactly $N$ rows. You can always remove the first column and that will leave you with $X - N$ boxes in $\leq N$ rows, i.e. a partition of $X - N$ with at most $N$ rows.
For example, if you want to enumerate the partitions of $9$ into $3$ parts, you would consider all partitions of $6$ with at most $3$ parts. This gives $6,51,42,411,33,321,222$ (assuming I haven't missed any), so the partitions of $9$ into exactly $3$ parts, obtained by adding $1$ to each part including "$0$ parts" if there are any, are $711,621,531,522,441,432,333$.
Note that if $N \geq X/2$, then any partition of $X - N$ will have $\leq N$ parts, so in that case the number of partitions of $X$ into $N$ parts will be exactly the number of partitions of $X - N$.