Permutations with forbidden subsequences and nonseparable planar maps
FPSAC '93 Proceedings of the 5th conference on Formal power series and algebraic combinatorics
Journal of Combinatorial Theory Series A
A combinatorial proof of J. West's conjecture
Discrete Mathematics
Random maps, coalescing saddles, singularity analysis, and airy phenomena
Random Structures & Algorithms - Special issue on analysis of algorithms dedicated to Don Knuth on the occasion of his (100)8th birthday
Description trees and Tutte formulas
Theoretical Computer Science
Analytic Combinatorics
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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, @b(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 @b(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 Steingrimsson 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.