Stack Sorting with Increasing and Decreasing Stacks

10/08/2019

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by   Giulio Cerbai, et al.

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We introduce a sorting machine consisting of k+1 stacks in series: the first k stacks can only contain elements in decreasing order from top to bottom, while the last one has the opposite restriction. This device generalizes <cit.>, which studies the case k=1. Here we show that, for k=2, the set of sortable permutations is a class with infinite basis, by explicitly finding an antichain of minimal nonsortable permutations. This construction can easily be adapted to each k > 3. Next we describe an optimal sorting algorithm, again for the case k=2. We then analyze two types of left-greedy sorting procedures, obtaining complete results in one case and only some partial results in the other one. We close the paper by discussing a few open questions.