On linearization coefficients of \(q\)-Laguerre polynomials (Q5918846)

Summary: 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.