A construction of integer-valued polynomials with prescribed sets of lengths of factorizations

For an arbitrary finite non-empty set S of natural numbers greater 1, we
construct f ∈ Int(Z) = {g ∈ Q[x] | g(Z) ⊆ Z} such that S is the set of lengths of
f , i.e., the set of all n such that f has a factorization as a product of n irreducibles
in Int(Z). More generally, we can realize any finite non-empty multi-set of natural
numbers greater 1 as the multi-set of lengths of the essentially different factorizations of f