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Séminaire Lotharingien de Combinatoire, 84B.55 (2020), 12 pp.

# Byung-Hak Hwang, Jang Soo Kim, Jaeseong Oh and Sang-Hoon Yu

# On Linearization Coefficients of *q*-Laguerre Polynomials

**Abstract.**
The linearization coefficient **L**(*L*_{n1}(*x*) ... *L*_{nk}(*x*)) of
classical Laguerre polynomials *L*_{n}(*x*) is well known to be equal to the
number of (*n*_{1},...,*n*_{k})-derangements, which are permutations with a
certain condition. Kasraoui, Zeng and Stanton 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},...,*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.

Received: November 20, 2019.
Accepted: February 20, 2020.
Final version: April 30, 2020.

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