Factorization of integer-valued polynomials with square-free denominator

We describe an algorithm to compute the essentially different factorizations of a given image primitive integer-valued polynomial $f(X)=g(X)\mathrm{/}d\in \text{Q}[X]$, where $g\in \mathbb{Z}[X]$ and $d\in \mathbb{N}$ is square-free, assuming that the factorization of $g(X)$ in $\mathbb{Z}[X]$ and $d$ in $\mathbb{Z}$ is known. We translate this problem into a combinatorial one.