Counting Separable Polynomials in $\mathbb{Z}\mathrm{/}n[x]$

For a commutative ring $R$, a polynomial $f\in R[x]$ is called separable if $R[x]\mathrm{/}f$ is a separable $R$-algebra. We derive formulae for the number of separable polynomials when $R=\mathbb{Z}\mathrm{/}n$, extending a result of L. Carlitz. For instance, we show that the number of separable polynomials in $\mathbb{Z}\mathrm{/}n[x]$ that are separable is $\varphi (n)n^d{\prod }_i(1-p_i^{-d})$ where $n=\prod p_i^{k_i}$ is the prime factorisation of $n$ and $\varphi$ is Euler’s totient function.