  We obtain estimates on the number $|\mathcal{A}_{\boldsymbol{\lambda}}|$ of
elements on a linear family $\mathcal{A}$ of monic polynomials of
$\mathbb{F}_q[T]$ of degree $n$ having factorization pattern
$\boldsymbol{\lambda}:=1^{\lambda_1}2^{\lambda_2}\cdots n^{\lambda_n}$. We show
that $|\mathcal{A}_{\boldsymbol{\lambda}}|=
\mathcal{T}(\boldsymbol{\lambda})\,q^{n-m}+\mathcal{O}(q^{n-m-{1}/{2}})$, where
$\mathcal{T}(\boldsymbol{\lambda})$ is the proportion of elements of the
symmetric group of $n$ elements with cycle pattern $\boldsymbol{\lambda}$ and
$m$ is the codimension of $\mathcal{A}$. Furthermore, if the family
$\mathcal{A}$ under consideration is "sparse", then
$|\mathcal{A}_{\boldsymbol{\lambda}}|=
\mathcal{T}(\boldsymbol{\lambda})\,q^{n-m}+\mathcal{O}(q^{n-m-{1}})$. Our
estimates hold for fields $\mathbb{F}_q$ of characteristic greater than 2. We
provide explicit upper bounds for the constants underlying the
$\mathcal{O}$--notation in terms of $\boldsymbol{\lambda}$ and $\mathcal{A}$
with "good" behavior. Our approach reduces the question to estimate the number
of $\mathbb{F}_q$--rational points of certain families of complete
intersections defined over $\mathbb{F}_q$. Such complete intersections are
defined by polynomials which are invariant under the action of the symmetric
group of permutations of the coordinates. This allows us to obtain critical
information concerning their singular locus, from which precise estimates on
their number of $\mathbb{F}_q$--rational points are established.
