  Let $D$ be a Dedekind domain with infinitely many maximal ideals, all of
finite index, and $K$ its quotient field. Let $\operatorname{Int}(D) = \{f\in
K[x] \mid f(D) \subseteq D\}$ 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 $\operatorname{Int}(D)$ which has exactly $n$
essentially different factorizations into irreducibles in
$\operatorname{Int}(D)$, 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 $\operatorname{Int}(D)$ to a block monoid.
