Stack-sorting for Words

TL;DR

This paper introduces two new stack-sorting operators for words, analyzes their properties, and connects sorting problems to pattern avoidance, providing combinatorial formulas and open questions.

Contribution

It generalizes stack-sorting operators to words, develops methods to count preimages, and relates sorting to pattern avoidance with new enumerative results.

Findings

The operator 'tortoise' sorts words faster than 'hare'.

Derived recurrence for sortable words with fixed letter multiplicities.

Counted pattern-avoiding words using Catalan numbers.

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.