  The primary goal of this document is to record the asymptotic effects that
preimage constraints impose upon the sizes of the iterated images of a random
function. Specifically, given a subset $\mathcal{P}\subseteq \mathbb{Z}_{\geq
0}$ and a finite set $S$ of size $n$, choose a function uniformly from the set
of functions $f:S\rightarrow S$ that satisfy the condition that
$|f^{-1}(x)|\in\mathcal{P}$ for each $x\in S$, and ask what $|f^k(S)|$ looks
like as $n$ goes to infinity. The robust theory of singularity analysis allows
one to completely answer this question if one accepts that $0\in\mathcal{P}$,
that $\mathcal{P}$ contains an element bigger than 1, and that
$\gcd(\mathcal{P})=1$; only the third of these conditions is a meaningful
restriction. The secondary goal of this paper is to record much of the
background necessary to achieve the primary goal.
