  The linearization coefficient $\mathcal{L}(L_{n_1}(x)\dots L_{n_k}(x))$ of
classical Laguerre polynomials $L_n(x)$ is known to be equal to the number of
$(n_1,\dots,n_k)$-derangements, which are permutations with a certain
condition. Kasraoui, Stanton and Zeng found a $q$-analog of this result using
$q$-Laguerre polynomials with two parameters $q$ and $y$. Their formula
expresses the linearization coefficient of $q$-Laguerre polynomials as the
generating function for $(n_1,\dots,n_k)$-derangements with two statistics
counting weak excedances and crossings. In this paper their result is proved by
constructing a sign-reversing involution on marked perfect matchings.

Combinatorics