A new combinatorial interpretation of a $q$-analogue of the Lah numbers

Abstract

The Lah numbers L(n, k) are the connection constants between the rising factorial and falling factorial polynomial bases and count partitions of n distinct objects into k blocks, where objects within a block are ordered (termed Laguerre configurations). In this paper, we consider the q-Lah numbers defined as the connection constants between the comparable bases of polynomials gotten by replacing each positive integer n with nq = 1+ q + · · ·+ q n−1 and provide a new combinatorial interpretation for them by describing a statistic on Laguerre configurations for which they are the generating function. We 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.