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

TL;DR

This paper introduces two new generalizations of Lah numbers, explores their properties, and establishes connections between them and existing combinatorial and algebraic structures.

Contribution

It defines Lah numbers with higher level and Lah numbers of order s, providing combinatorial and algebraic frameworks and linking these generalizations.

Findings

Established a connection between Lah numbers with higher level and (l,r)-Lah numbers.

Derived properties of Lah numbers of order s and Lah polynomials of order s.

Proved relationships between the two introduced generalizations.

Abstract

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.