The Lah Numbers with Higher Level and the Lah Numbers of Order s

In this paper we introduce and study two generalizations of Lah numbers, analogous to the Stirling numbers with higher level - a combinatorial one (Lah numbers with higher level) and an algebraic one (Lah numbers of order $s$). We define the Lah numbers with higher level following a combinatorial approach and the Lah numbers of order $s$ following an algebraic approach. We prove a direct connection between the Lah numbers with higher level and the $(l,r)$-Lah numbers. Some properties of the Lah numbers of order $s$ and Lah polynomials of order $s$ are given. Finally, we prove connections between these two generalizations.