# Restricted non-separable planar maps and some pattern avoiding permutations

### Discrete Applied Mathematics, Volume 161, Issues 16–17, (2013), Pages 2514-2526

Sergey, Pavel, Christopher and Henning

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.

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## Presentations

- Kort, tré og mynstur, Conference of the Icelandic Mathematics Society 2011 (Henning presented)
- Restricted rooted non-separable planar maps, Joint Mathematics Meetings, Boston, MA, January 2012 (Henning presented)
- Crazy bijections between planar maps, beta-trees and perms, ICE-TCS seminar, Reykjavik University, March 2013 (Henning presented)