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The subject of pattern avoiding permutations has its roots in computerscience, namely in the problem of sorting a permutation through a stack. Aformula for the number of permutations of length n that can be sorted bypassing it twice through a stack (where the letters on the stack have to be inincreasing order) was conjectured by West, and later proved by Zeilberger.Goulden and West found a bijection from such permutations to nonseparableplanar maps, and later, Jacquard and Schaeffer presented a bijection from theseplanar maps to certain labeled plane trees, called beta(1,0)-trees. Usinggenerating trees, Dulucq, Gire and West showed that nonseparable planar mapsare equinumerous with permutations avoiding the (classical) pattern 2413 andthe barred pattern 41\bar{3}52; they called these permutations nonseparable. Wegive a new bijection between beta(1,0)-trees and permutations avoiding thedashed patterns 3-1-4-2 and 2-41-3. These permutations can be seen to beexactly the reverse of nonseparable permutations. Our bijection is built usingdecompositions of the permutations and the trees, and it translates sevenstatistics on the trees into statistics on the permutations. Among thestatistics involved are ascents, left-to-right minima and right-to-left maximafor the permutations, and leaves and the rightmost and leftmost paths for thetrees. In connection with this we give a nontrivial involution on thebeta(1,0)-trees, which specializes to an involution on unlabeled rooted planetrees, where it yields interesting results. Lastly, we conjecture the existenceof a bijection between nonseparable permutations and two-stack sortablepermutations preserving at least four permutation statistics.

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