Contents Online
Journal of Combinatorics
Volume 2 (2011)
Number 2
A new combinatorial interpretation of a $q$-analogue of the Lah numbers
Pages: 245 – 264
DOI: https://dx.doi.org/10.4310/JOC.2011.v2.n2.a4
Authors
Abstract
The Lah numbers $L(n,k)$ are the connection constants between therising factorial and falling factorial polynomial bases and countpartitions of $n$ distinct objects into $k$ blocks, where objectswithin a block are ordered (termed $Laguerre configurations$).In this paper, we consider the $q$-Lah numbersdefined as the connection constants between the comparable basesof polynomials obtained by replacing each positive integer $n$ with$n_{q}=1+q+\cdots+q^{n-1}$ and provide a new combinatorial interpretationfor these numbers by describing a statistic on Laguerre configurations forwhich they are the generating function. We describe someother algebraic properties of these numbers and can provide combinatorialexplanations in several instances using our interpretation.A further generalization involving a second parameter may also be given.
Keywords
Laguerre configuration, Lah numbers, q-analogue, statistic
2010 Mathematics Subject Classification
Primary 05A15. Secondary 05A05.
Published 25 October 2011