Polynomial functions of the ring $\mathbb{Z}_m[\alpha]$ of dual numbers modulo $m$

For any ring $R$, the dual numbers ring over $R$ is $R[\alpha] = R[x]/(x^2)$. We characterize when a polynomial $f\in R[\alpha][x]$ is a null polynomial or a permutation polynomial on $R[\alpha]$ in terms of the behaviour of its coordinate polynomials $f_1,f_2\in R[x]$, where $f=f_1+f_2\alpha$, on $R$. By means of determining the order of the pointwise stabilizer of the subring of constants $\mathbb{Z}_{p^n}$ in the group of polynomial permutations of $\mathbb{Z}_{p^n}[\alpha]$ we derive explicit formulas for the number of polynomial functions and polynomial permutations on $\mathbb{Z}_{p^n}[\alpha]$ for $n\le p$.