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.