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arXiv · 1809.09158

Stack-sorting for Words

Colin Defant·Noah Kravitz
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Abstract

We introduce operators $\mathsf{hare}$ and $\mathsf{tortoise}$, which act on words as natural generalizations of West's stack-sorting map. We show that the heuristically slower algorithm $\mathsf{tortoise}$ can sort words arbitrarily faster than its counterpart $\mathsf{hare}$. We then generalize the combinatorial objects known as valid hook configurations in order to find a method for computing the number of preimages of any word under these two operators. We relate the question of determining which words are sortable by $\mathsf{hare}$ and $\mathsf{tortoise}$ to more classical problems in pattern avoidance, and we derive a recurrence for the number of words with a fixed number of copies of each letter (permutations of a multiset) that are sortable by each map. In particular, we use generating trees to prove that the $\ell$-uniform words on the alphabet $[n]$ that avoid the patterns $231$ and $221$ are counted by the $(\ell+1)$-Catalan number $\frac{1}{\ell n+1}{(\ell+1)n\choose n}$. We conclude with several open problems and conjectures.

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Colin Defant, Noah Kravitz. 2018-09-24. Stack-sorting for Words. https://arxiv.org/abs/1809.09158

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