The number of maximal period polynomial mappings over the Galois fields of odd characteristics

Let $R = GR(q^n, p^n)$ be a Galois ring of cardinality $q^n$ and characteristics $p^n$, where $q = p^m$, $m, n > 1$. Let the sequence $U = \{u_i\}$ is defined by equations $u_{i+1} = f(u_i)$, $i \in \mathbb N_0$, and $f$ be a polynomial mapping of the ring $R$. It was proved earlier that the maximal possible period of $U$ equals $q(q-1)p^{n-2}$. Here we find the number of polynomial mappings over $R$ having maximal possible periods for $p\ne2$.