If I understand correctly, you are counting the conjugacy classes of elements of order $p$ in $\operatorname{GL}(n,\mathbb{F}_p) $ — or equivalently, of matrices $N$ in $\operatorname{M}_n(\mathbb{F}_p) $ with $N^p=0$, but $N\neq 0$. If $p\geq n$ you get all nilpotent matrices, hence indeed $P(n)-1$ conjugacy classes using Jordan normal form, but this is false for $p<n$: for $p=2$, for instance, you get $[n/2]$ (corresponding to partitions with only 1 and 2, and at least one 2).

If I understand correctly, you are counting the conjugacy classes of elements of order $p$ in $\operatorname{GL}(n,\mathbb{F}_p) $ — or equivalently, of matrices $N$ in $\operatorname{M}_n(\mathbb{F}_p) $ with $N^p=0$, but $N\neq 0$. If $p\geq n$ you get all nilpotent nonzero matrices, hence indeed $P(n)-1$ conjugacy classes using Jordan normal form, but this is false for $p<n$: for $p=2$, for instance, you get $[n/2]$ (corresponding to partitions with only 1 and 2, and at least one 2).