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 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.