On the combinatorics of quadrant marked mesh patterns in 132-avoiding permutations

International audienceThe 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 quadrant marked mesh patterns where we specify conditions on exactly one of the four possible quadrants in a quadrant marked mesh pattern. We show that for the first two quadrants, certain of these coefficients are counted by elements of Catalan's triangle and give a new combinatorial interpretation of these coefficients for quadrant four. We also give a new bijection between 132-avoiding permutations and non-decreasing parking functions

Distributions of several infinite families of mesh patterns

Br\"and\'en and Claesson introduced mesh patterns to provide explicit expansions for certain permutation statistics as linear combinations of (classical) permutation patterns. The first systematic study of avoidance of mesh patterns was conducted by Hilmarsson et al., while the first systematic study of the distribution of mesh patterns was conducted by the first two authors. In this paper, we provide far-reaching generalizations for 8 known distribution results and 5 known avoidance results related to mesh patterns by giving distribution or avoidance formulas for certain infinite families of mesh patterns in terms of distribution or avoidance formulas for smaller patterns. Moreover, as a corollary to a general result, we find the distribution of one more mesh pattern of length 2.Comment: 27 page

Equidistributions of mesh patterns of length two and Kitaev and Zhang's conjectures

A systematic study of avoidance of mesh patterns of length 2 was conducted by Hilmarsson et al. in 2015. In a recent paper Kitaev and Zhang examined the distribution of the aforementioned patterns. The aim of this paper is to prove more equidistributions of mesh pattern and confirm Kitaev and Zhang's four conjectures by constructing two involutions on permutations.Comment: 15 pages,3 figures, comments are welcom

Parking Functions on Trees and Directed Graphs

A parking function can be thought of as a sequence of n drivers, each with a preferred parking space, wanting to park along a one-way street with n parking spaces. Each driver checks her preferred parking space and, if it is occupied, parks in the first available space afterwards. One may consider how the enumeration of these sequences changes if the “parking lot” is made more complex, a question whose solution this dissertation lays the foundations for and answers in the case of certain families of graphs. We begin by generalizing the underlying parking lot to a general digraph and give several equivalent characterizations. We then start by building on recent work in the case that the parking lot is a tree with edges directed towards a root. We generalize the notions of “prime” and “increasing” parking functions to give enumerative results concerning both. Additionally, we consider one of the numerous statistics on classical parking functions, the number of drivers who park in their desired spot, and show that it connects these tree parking functions that are prime and those that are both increasing and prime. Finally, we consider miscellaneous results on various families of trees and enumerate a generalization of Dyck paths in the proces