A new combinatorial interpretation of a \(q\)-analogue of the Lah numbers (Q433477)

The \textit{Lah numbers} \(L(n,k)\) [\textit{I. Lah}, ``Eine neue Art von Zahlen, ihre Eigenschaften und Anwendung in der mathematischen Statistik,'' Mitt.-Bl.\ Math.\ Statistik 7, 203--212 (1955; Zbl 0066.11801)] were introduced as the ``connection constants'' in the polynomial identities NEWLINE\[NEWLINEx(x+1)\dots(x+n-1)=\sum^n_{k=0}L(n,k)\cdot x(x-1)\dots(x-k+1)\;\qquad\forall n\in\mathbb{N}.NEWLINE\]NEWLINE The \(q\)-Lah numbers \(L_q(n,k)\) are defined as the ``connection constants'' in the identities NEWLINE\[NEWLINEx\left(x+1_q\right)\dots\left(x+(n-1)_q\right)=\sum^n_{k=0}L_q(n,k)\cdot x\left(x-1_q\right)\dots\left(x-(k-1)_q\right)\;\qquad\forall n\in\mathbb{N},NEWLINE\]NEWLINE where \(n_q\) is defined to be the polynomial \(1+q+\dots+q^{n-1}\) for positive integers \(n\), and an indeterminate \(q\); \(0_q=0\). A \textit{Laguerre configuration} is a distribution of labelled objects into unlabelled contents-ordered boxes with no box left empty [\textit{A.M. Garsia} and \textit{J. Remmel}, ``A combinatorial interpretation of \(q\)-derangement and \(q\)-Laguerre numbers,'' Eur.\ J.\ Comb.\ 1, 47--59 (1980; Zbl 0462.05012)].NEWLINENEWLINEFrom the authors' abstract: (The authors) ``provide a new combinatorial interpretation for (the numbers \(n_q\)) by describing a statistic on Laguerre configurations for which they are the generating function. (They) describe some other algebraic properties of these numbers and can provide combinatorial explanations in several instances using our interpretation. A further generalization involving a second parameter may also be given.''