Quadrant Marked Mesh Patterns and the r-Stirling Numbers

Matt Davis
2015 Journal of Integer Sequences   unpublished
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
more » ... d by Kitaev and Remmel and show a connection to another mesh pattern introduced by Kitaev and Liese.