Geodesic
Quadrant marked mesh patterns and the \(r\)-Stirling numbers
Journal of integer sequences, Tome 18 (2015) no. 10
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.
Classification : 05A15, 05E15
Keywords: permutation pattern, marked mesh pattern, permutation statistic, r-Stirling number, harmonic number 
@article{JIS_2015__18_10_a2,
     author = {Davis,  Matt},
     title = {Quadrant marked mesh patterns and the {\(r\)-Stirling} numbers},
     journal = {Journal of integer sequences},
     year = {2015},
     volume = {18},
     number = {10},
     zbl = {1328.05007},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2015__18_10_a2/}
}
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Davis,  Matt. Quadrant marked mesh patterns and the \(r\)-Stirling numbers. Journal of integer sequences, Tome 18 (2015) no. 10. http://geodesic.mathdoc.fr/item/JIS_2015__18_10_a2/