Quadrant Marked Mesh Patterns
نویسندگان
چکیده
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.
منابع مشابه
On the combinatorics of quadrant marked mesh patterns in 132-avoiding permutations
The study of quadrant marked mesh patterns in 132-avoiding permutations was initiated by Kitaev, Remmel and Tiefenbruck. We refine several results of Kitaev, Remmel and Tiefenbruck by giving new combinatorial interpretations for the coefficients that appear in the generating functions for the distribution of quadrant marked mesh patterns in 132-avoiding permutations. In particular, we study qua...
متن کاملQuadrant Marked Mesh Patterns in Alternating Permutations
Abstract. This paper is a continuation of the systematic study of the distribution of quadrant marked mesh patterns initiated by the authors in [J. Integer Sequences 12 (2012), Article 12.4.7]. We study quadrant marked mesh patterns on up-down and down-up permutations, also known as alternating and reverse alternating permutations, respectively. In particular, we refine classical enumeration re...
متن کاملQuadrant Marked Mesh Patterns and the r-Stirling Numbers
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 numb...
متن کاملThe Parking Problem for Finite-State Robots
This paper is a step toward understanding the algorithmic concomitants of modeling robots as mobile finite-state machines (FSMs, for short) that travel within square two-dimensional meshes (which abstract the floors of laboratories or factories or warehouses). We study the ability of FSMs to scalably perform a simple path-planning task called parking, within fixed square meshes of arbitrary siz...
متن کاملLaparoscopic Evaluation of Abdominal Adhesions With Different Prosthetic Meshes in Rabbits
BACKGROUND The use of prosthetic materials to reinforce the abdominal wall is associated with a low index of recurrence; however, intraperitoneal placement of a foreign body may lead to adhesions. The present investigation was designed to determine adhesion formation with commercially available meshes implanted laparoscopically in rabbits. METHODS Three different meshes were implanted laparos...
متن کامل