Restricted non-separable planar maps and some pattern avoiding permutations

Tutte founded the theory of enumeration of planar maps in a series of papers in the 1960s. Rooted non-separable planar maps are in bijection with West-2-stack-sortable permutations, β(1,0)-trees introduced by Cori, Jacquard and Schaeffer in 1997, as well as a family of permutations defined by the avoidance of two four letter patterns. In this paper we study how certain structures in planar maps transfer to trees and permutations via the bijections. More precisely, we show that the number of 2-faces in a map equals the number of nodes in the corresponding β(1,0)-tree that are single children with maximum label; give upper and lower bounds on the number of multiple-edge-free rooted non-separable planar maps. We also use the bijection between rooted non-separable planar maps and a certain class of permutations, found by Claesson, Kitaev and Steingrímsson in 2009, to show that 2-face-free maps correspond to permutations avoiding certain mesh patterns. Finally, we give asymptotics for some of our enumerative results.