Journal of Integer Sequences, Vol. 18 (2015), Article 15.10.1

Quadrant Marked Mesh Patterns and the r-Stirling Numbers

Matt Davis
Department of Mathematics and Computer Science
Muskingum University
163 Stormont St.
New Concord, OH 43762


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.

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(Concerned with sequences A001712 A052881.)

Received November 30 2014; revised versions received April 12 2015; August 31 2015. Published in Journal of Integer Sequences, September 8 2015.

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