Decompositions and statistics for beta(1,0)-trees and nonseparable permutations

TL;DR

This paper introduces a new bijection between beta(1,0)-trees and certain pattern-avoiding permutations, revealing structural insights and statistical correspondences, and proposes conjectures on two-stack sortable permutations.

Contribution

It presents a novel bijection linking beta(1,0)-trees with specific pattern-avoiding permutations and explores associated permutation and tree statistics.

Findings

Bijection between beta(1,0)-trees and permutations avoiding 3-1-4-2 and 2-41-3 patterns

Statistics on trees are translated into permutation statistics via the bijection

An involution on beta(1,0)-trees yields new combinatorial results

Abstract

The subject of pattern avoiding permutations has its roots in computer science, namely in the problem of sorting a permutation through a stack. A formula for the number of permutations of length n that can be sorted by passing it twice through a stack (where the letters on the stack have to be in increasing order) was conjectured by West, and later proved by Zeilberger. Goulden and West found a bijection from such permutations to nonseparable planar maps, and later, Jacquard and Schaeffer presented a bijection from these planar maps to certain labeled plane trees, called beta(1,0)-trees. Using generating trees, Dulucq, Gire and West showed that nonseparable planar maps are equinumerous with permutations avoiding the (classical) pattern 2413 and the barred pattern 41\bar{3}52; they called these permutations nonseparable. We give a new bijection between beta(1,0)-trees and permutations avoiding the dashed patterns 3-1-4-2 and 2-41-3. These permutations can be seen to be exactly the reverse of nonseparable permutations. Our bijection is built using decompositions of the permutations and the trees, and it translates seven statistics on the trees into statistics on the permutations. Among the statistics involved are ascents, left-to-right minima and right-to-left maxima for the permutations, and leaves and the rightmost and leftmost paths for the trees. In connection with this we give a nontrivial involution on the beta(1,0)-trees, which specializes to an involution on unlabeled rooted plane trees, where it yields interesting results. Lastly, we conjecture the existence of a bijection between nonseparable permutations and two-stack sortable permutations preserving at least four permutation statistics.