Subword enumeration up to stack-sorting equivalence

2026-04-28

AI Summary

This work extends the West stack-sorting map from permutations to finite words and employs it to investigate structural properties of infinite words. Building upon the Defant–Kravitz generalization of stack sorting, the authors introduce tortoise and hare rearrangement operations to define a novel subword enumeration scheme and generalize the abelian complexity function. They establish, for the first time in combinatorics on words, a stack-sorting equivalence relation, thereby forging a bridge between permutation stack sorting and the study of infinite word structure. The framework is successfully applied to paperfolding and Thue–Morse words, revealing new connections between special factors of the Thue–Morse word and its stack-sorting behavior, thus highlighting distinctive structural features inherent to this classical automatic sequence.

Abstract

Defant and Kravitz introduced generalizations of West's stack-sorting map $s$ from permutations to finite words. This raises questions as to how such generalizations could be applied in the field of combinatorics on words. The Defant-Kravitz generalizations of $s$ depend on how repeated occurrences of the same character within a word may be repositioned, according to their $\textsf{tortoise}$ and $\textsf{hare}$ operations. As demonstrated in this paper, these operations provide a natural way of extending abelian complexity functions for infinite sequences, in a way that gives light to structural properties associated with infinite words. We apply these new ideas to two famous infinite words: the paperfolding word and the Thue-Morse word. In the case of the Thue-Morse word, we discover an interesting connection to the previous work of several authors, such as de Luca and Varricchio, on the ``special'' factors of the Thue-Morse word. This may be seen as providing a basis for a new and interdisciplinary area linking the combinatorics about the stack-sorting of permutations with the field of combinatorics on words.

Problem

• stack-sorting
• combinatorics on words
• infinite words
• Thue-Morse word
• paperfolding word

Innovation

• stack-sorting
• combinatorics on words
• abelian complexity
• Thue-Morse word
• subword enumeration