Sets of lengths of factorizations of in...

Let $D$ be a Dedekind domain with infinitely many maximal ideals, all of finite index, and $K$ its quotient field.  Let $\Int(D) = \{f\in K[x] \mid f(D) \subseteq D\}$ be the ring of integer-valued polynomials on $D$.

Given any finite multiset $\{k_1, \ldots, k_n\}$ of integers greater than $1$, we construct a polynomial in $\Int(D)$ which has exactly $n$ essentially different factorizations into irreducibles in $\Int(D)$, the lengths of these factorizations being $k_1$, \ldots, $k_n$.  We also show that there is no transfer homomorphism from the multiplicative monoid of $\Int(D)$ to a block monoid.