Stack Sorting, Trees, and Pattern Avoidance

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 certain planar maps, and later Cori, Jacquard and Schaeffer presented a bijection from these planar maps to certain labeled plane trees, called β(1, 0)-trees. We show that these labeled plane trees are in one-to-one correspondence with permutations that avoid the generalized patterns 3-1-4-2 and 2-41-3. We do this by establishing a bijection between the avoiders and the trees. This bijection translates 7 statistics on the trees into statistics on the avoiders. Among the statistics involved are ascents, left-to-right minima and right-toleft maxima for the permutations, and leaves and the rightmost and leftmost paths for the trees. Moreover, extensive computations of statistics on our avoiders, two-stack sortable permutations and the β(1, 0)-trees suggest that the avoiders are structurally more closely connected to the β(1, 0)-trees—and thus to the planar maps—than two-stack sortable permutations are. In connection with this we give a nontrivial involution on the β(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 avoiders and twostack sortable permutations preserving at least four permutation statistics.