Restricted rooted non-separable planar maps

Tutte founded the theory of enumeration of planar maps in a series of papers in the 1960s. Rooted non-separable planar maps have connections, for example, to pattern-avoiding permutations, and they are in one-to-one correspondence with the β(1, 0)-trees introduced by Cori, Jacquard and Schaeffer in 1997. In this paper we enumerate 2-face-free rooted non-separable planar maps and obtain restrictions on β(1, 0)-trees giving k-face-free rooted non-separable planar maps. Moreover, we discuss multiple-edge-free rooted non-separable planar maps and provide some rough lower bounds for their number using restricted β(1, 0)-trees. We enumerate so-called primitive rooted non-separable planar maps (which form a basis for generating all rooted non-separable planar maps). Finally, we discuss some equinumerous objects such as primitive β(1, 0)-trees and certain pattern-avoiding permutations.