Restricted rooted non-separable planar maps

Details Restricted rooted non-separable planar maps Restricted rooted non-separable planar maps

2012-02-08

arxiv.org/abs/1202.1790v1

Mathematics

Combinatorics

15 pages, 13 figures

Scientific paper

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 beta(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 beta(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 beta(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 beta(1,0)-trees and certain pattern-avoiding permutations.

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