Journal of Integer Sequences, Vol. 15 (2012), Article 12.4.7

Quadrant Marked Mesh Patterns

Sergey Kitaev
University of Strathclyde
Livingstone Tower, 26 Richmond Street
Glasgow G1 1XH
United Kingdom

Jeffrey Remmel
Department of Mathematics
University of California, San Diego
La Jolla, CA 92093-0112


In this paper we begin the first systematic study of distributions of quadrant marked mesh patterns. Mesh patterns were introduced recently by Brändén and Claesson in connection with permutation statistics. Quadrant marked mesh patterns are based on how many elements lie in various quadrants of the graph of a permutation relative to the coordinate system centered at one of the points in the graph of the permutation. We study the distribution of several quadrant marked mesh patterns in a symmetric group and in certain subsets of the symmetric group. We find explicit formulas for the generating function of such distributions in several general cases and develop recursions to compute the numbers in question in other cases. In addition, certain q-analogues of our results are discussed.

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(Concerned with sequences A000254 A000399 A000774 A000984 A001712 A006318 A007051 A128652.)

Received January 4 2012; revised version received April 1 2012. Published in Journal of Integer Sequences, April 11 2012.

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