Quadrant Marked Mesh Patterns and the r-Stirling Numbers

TL;DR

This paper explores a specific marked mesh pattern in permutations, linking its generating function to r-Stirling numbers, and derives new formulas connecting these to classical Stirling and harmonic numbers.

Contribution

It establishes a novel connection between a marked mesh pattern and r-Stirling numbers, providing new formulas and answering open questions in the field.

Findings

Generated function described by r-Stirling numbers.

Derived formulas linking r-Stirling, classical Stirling, and harmonic numbers.

Connected the pattern to another mesh pattern by Kitaev and Liese.

Abstract

Marked mesh patterns are a very general type of permutation pattern. We examine a particular marked mesh pattern originally defined by Kitaev and Remmel, and show that its generating function is described by the $r$-Stirling numbers. We examine some ramifications of various properties of the $r$-Stirling numbers for this generating function, and find (seemingly new) formulas for the $r$-Stirling numbers in terms of the classical Stirling numbers and harmonic numbers. We also answer some questions posed by Kitaev and Remmel and show a connection to another mesh pattern introduced by Kitaev and Liese.