Restricted non-separable planar maps and some pattern avoiding permutations

TL;DR

This paper explores the enumeration and structural properties of rooted non-separable planar maps, establishing connections with pattern-avoiding permutations and beta(1,0)-trees, and providing bounds and asymptotic results.

Contribution

It introduces bounds on multiple-edge-free maps, links map features to permutation patterns, and derives asymptotic enumerations, advancing understanding of these combinatorial structures.

Findings

Bounds on the number of multiple-edge-free rooted non-separable planar maps

Equivalence between the number of 2-faces and occurrences of a mesh pattern in permutations

Asymptotic formulas for enumeration of these maps and related structures

Abstract

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, beta(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 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 Steingrimsson in 2009, to show that the number of 2-faces (excluding the root-face) in a map equals the number of occurrences of a certain mesh pattern in the permutations. We further show that this number is also the number of nodes in the corresponding beta(1,0)-tree that are single children with maximum label. Finally, we give asymptotics for some of our enumerative results.