  We estimate the number $|\mathcal{A}_{\boldsymbol\lambda}|$ of elements on a
nonlinear family $\mathcal{A}$ of monic polynomials of $\mathbb{F}_q[T]$ of
degree $r$ having factorization pattern
$\boldsymbol\lambda:=1^{\lambda_1}2^{\lambda_2}\cdots r^{\lambda_r}$. We show
that $|\mathcal{A}_{\boldsymbol\lambda}|=
\mathcal{T}(\boldsymbol\lambda)\,q^{r-m}+\mathcal{O}(q^{r-m-{1}/{2}})$, where
$\mathcal{T}(\boldsymbol\lambda)$ is the proportion of elements of the
symmetric group of $r$ elements with cycle pattern $\boldsymbol\lambda$ and $m$
is the codimension of $\mathcal{A}$. We provide explicit upper bounds for the
constants underlying the $\mathcal{O}$--notation in terms of
$\boldsymbol\lambda$ and $\mathcal{A}$ with "good" behavior. We also apply
these results to analyze the average--case complexity of the classical
factorization algorithm restricted to $\mathcal{A}$, showing that it behaves as
good as in the general case.
