A Combinatorial Interpretation of q-Derangement and q-Laguerre Numbers

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Abstract

It is becoming increasingly clear from the recent works of the Lotharingianschool of combinatorics that special functions and identities of classicalmathematics are full of combinatorial information, which can be expressedin the form of "encodings" of objects into words of certain languages andnatural bijections between different classes of languages. The classicalidentities appear as relations between enumerators of these words bysuitable statistics. The authors have made a systematic study of thissubject leading to a rich inventory of correspondences that has led to newidentities as well as more revealing proofs of old ones. In particular theyhave studied the $q$-analogue of $n!$ and the polynomials $F_n(q,x)$,$L_n(q,x)$ and their generalizations, where $F_n(1,x)$ is the enumerator ofpermutations by number of fixed points and $L_n(1,x)$ is the ordinary<b class="highlight">Laguerre</b> polynomial which can be interpreted as the enumerator ofplacements of $n$ distinguishable objects in undistinguishable boxes.

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Garsia, A. M., & Remmel, J. (1980). A Combinatorial Interpretation of q-Derangement and q-Laguerre Numbers. European Journal of Combinatorics, 1(1), 47–59. https://doi.org/10.1016/S0195-6698(80)80021-7

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