Cyclic and Linear Graph Partitions and Normal Ordering

TL;DR

This paper explores graph-based Stirling and Lah numbers, revealing their role as coefficients in normal ordering problems and deriving related identities, including q-analogues, thus advancing combinatorial algebra understanding.

Contribution

It introduces graphical Stirling and Lah numbers and demonstrates their appearance in normal ordering, extending previous work and deriving new identities including q-analogues.

Findings

Graphical Stirling and Lah numbers are coefficients in normal ordering.

Identities involving q-analogues of these numbers are established.

Connections between graph partitions and algebraic normal ordering are clarified.

Abstract

The Stirling number of a simple graph is the number of partitions of its vertex set into a specific number of non-empty independent sets. In 2015, Engbers et al. showed that the coefficients in the normal ordering of a word $w$ in the alphabet $\{x,D\}$ subject to the relation $Dx=xD+1$ are equal to the Stirling number of certain graphs constructed from $w$. In this paper, we introduce graphical versions of the Stirling numbers of the first kind and the Lah numbers and show how they occur as coefficients in other normal ordering settings. Identities involving their $q$-analogues are also obtained.