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

Aleks Žigon Tankosič

TL;DR

This work develops two parallel generalizations of Lah numbers: a combinatorial higher-level variant and an algebraic order-$s$ variant, and then unifies them within a broader $(s,r)$-Lah framework. It defines the Lah numbers with higher level and derives a recurrence for the associated polynomials $Q_n^{s}(x)$, along with explicit sums and special cases, highlighting their combinatorial origin and connection to $(s,1)$-Lah numbers. It then introduces the Lah numbers of order $s$, gives their recurrence, derives relations with rising and falling factorials of higher level, and defines the Lah polynomials of order $s$ with a row-generating interpretation. Finally, it establishes inequalities and structural relations between the two generalizations and the classical Lah numbers, showing how they interlock and reduce to the ordinary case when $s=1$; these results broaden the combinatorial and algebraic toolkit for partition-based Lah-type numbers.

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.

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

TL;DR

This work develops two parallel generalizations of Lah numbers: a combinatorial higher-level variant and an algebraic order- variant, and then unifies them within a broader -Lah framework. It defines the Lah numbers with higher level and derives a recurrence for the associated polynomials , along with explicit sums and special cases, highlighting their combinatorial origin and connection to -Lah numbers. It then introduces the Lah numbers of order , gives their recurrence, derives relations with rising and falling factorials of higher level, and defines the Lah polynomials of order with a row-generating interpretation. Finally, it establishes inequalities and structural relations between the two generalizations and the classical Lah numbers, showing how they interlock and reduce to the ordinary case when ; these results broaden the combinatorial and algebraic toolkit for partition-based Lah-type numbers.

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 ). We define the Lah numbers with higher level following a combinatorial approach and the Lah numbers of order following an algebraic approach. We prove a direct connection between the Lah numbers with higher level and the -Lah numbers. Some properties of the Lah numbers of order and Lah polynomials of order are given. Finally, we prove connections between these two generalizations.

Paper Structure

This paper contains 9 sections, 14 theorems, 97 equations, 1 figure, 6 tables.

Table of Contents

  1. Introduction
  2. The Lah Numbers with Higher Level
  3. Recurrence relation for polynomials $Q_{n}^{s}(x)$
  4. The Lah Numbers of Order $s$
  5. Definition, Recurrence Relation and Some Special Cases
  6. Relations via Falling and Rising Factorials with Higher Level
  7. Matrix representation
  8. The Lah Polynomials of Order $s$
  9. Connections Between the Lah Numbers with Higher Level and the Lah Numbers of Order $s$

Key Result

Theorem 2.3

The Lah number with higher level coincides with the $(s, 1)$-Lah number (or, equivalently $(s, 0)$-Lah number). That is for all $s \geq 1$,

Figures (1)

  • Figure 1: A diagram showing how Stirling numbers with higher level of both kinds and Lah numbers of order $s$ give coefficients for changing one basis of polynomials to another (the standard basis and the bases of falling and rising factorials with higher level).

Theorems & Definitions (40)

  • Remark 1.1
  • Definition 2.1
  • Remark 2.2
  • Example
  • Theorem 2.3
  • proof
  • Corollary 2.4
  • Theorem 2.5
  • proof
  • Remark 2.6
  • ...and 30 more