Further bijections to pattern-avoiding valid hook configurations

Valid hook configurations are combinatorial objects used to understand West's stack-sorting map. We extend existing bijections between valid hook configurations and intervals in partial orders on Motzkin paths. To enumerate valid hook configurations on 312-avoiding permutations, we build off of an existing bijection into a Motzkin poset and construct a bijection to certain well-studied closed lattice walks in the first quadrant. We use existing results about these lattice paths to show that valid hook configurations on 312-avoiding permutations are not counted by a D-finite generating function, resolving a question of Defant, and additionally to obtain an asymptotic estimate for the number of such configurations. We also extend a bijection of Defant to a correspondence between valid hook configurations on 132-avoiding permutations and intervals in the Motzkin-Tamari posets, providing a more elegant proof of Defant's enumeration thereof. To investigate this bijection, we present a number of lemmas about valid hook configurations that are generally applicable and further study the bijections of Defant.