Journal of Combinatorics

Volume 2 (2011)

Number 2

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

Pages: 245 – 264

DOI: http://dx.doi.org/10.4310/JOC.2011.v2.n2.a4

Authors

Jim Lindsay (Mathematics Department, University of Tennessee, Knoxville, Tenn., U.S.A.)

Toufik Mansour (Mathematics Department, University of Haifa, Israel)

Mark Shattuck (Mathematics Department, University of Tennessee, Knoxville, Tenn., U.S.A.)

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.

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