A simple combinatorial interpretation of certain generalized Bell and Stirling numbers

TL;DR

This paper presents two natural combinatorial interpretations for a broad class of generalized Bell and Stirling numbers, simplifying their understanding and linking them to graph colorings and Eulerian digraphs.

Contribution

It introduces two new combinatorial models for generalized Bell and Stirling numbers, making their interpretation more intuitive and accessible.

Findings

Two combinatorial interpretations: graph colorings and labeled Eulerian digraphs.

Simplification of the combinatorial understanding of generalized Bell and Stirling numbers.

Broader applicability to the Boson normal ordering problem.

Abstract

In a series of papers, P. Blasiak et al. developed a wide-ranging generalization of Bell numbers (and of Stirling numbers of the second kind) that appears to be relevant to the so-called Boson normal ordering problem. They provided a recurrence and, more recently, also offered a (fairly complex) combinatorial interpretation of these numbers. We show that by restricting the numbers somewhat (but still widely generalizing Bell and Stirling numbers), one can supply a much more natural combinatorial interpretation. In fact, we offer two different such interpretations, one in terms of graph colourings and another one in terms of certain labelled Eulerian digraphs.