Lah Matrix and Its Algebraic Properties

Abstract

The $n\times n$ Lah matrix $L_n$ is defined by $(L_n{)}_{ij}=l(i,j)$, where $l(i,j)$ is the unsigned Lah number. In this paper, we investigate the algebraic properties of $L_n$, and many important relations between $L_n$ and Pascal matrix and Stirling matrix, respectively. In addition, we obtain its exponential expansion and Pascal matrix factorization. Furthermore, we introduce a simple method to find and prove combinatorial identities.