More results on stack-sorting for set partitions

Let a sock be an element of an ordered finite alphabet A and a sequence of these elements be a sock sequence. In 2023, Xia introduced a deterministic version of Defant and Kravitz's stack-sorting map by defining the φσ and φσ̄ pattern-avoidance stack-sorting maps for sock sequences. Xia showed that the φaba map is the only one that eventually sorts all set partitions; in this paper, we prove deeper results regarding φaba and φabā as a natural next step. We newly define two algorithms with time complexity O(n³) that determine if any given sock sequence is in the image of φaba or φabā respectively. We also show that the maximum number of preimages that a sock sequence of length n has grows at least exponentially under both the φaba and φabā maps. Additionally, we prove results regarding fertility numbers (introduced by Defant) in the context of set partitions and multiple-pattern-avoiding stacks.