Raney Paths and a Combinatorial Relationship between Rooted Nonseparable Planar Maps and Two-Stack-Sortable Permutations

An encoding of the set of two-stack-sortable permutations (TSS) in terms of lattice paths and ordered lists of strings is obtained. These lattice paths are called Raney paths. The encoding yields combinatorial decompositions for two complementary subsets of TSS, which are the analogues of previously known decompositions for the set of nonseparable rooted planar maps (NS). This provides a combinatorial relationship between TSS and NS, and, hence, a bijection is determined between these sets that is different, simpler, and more refined than the previously known bijection.