Year: 2023
Type: article
Abstract: A sock sequence is a sequence of elements, which we will refer to as socks, from a finite alphabet.A sock sequence is sorted if all occurrences of a sock appear consecutively.We define equivalence classes of sock sequences called sock patterns, which are in bijection with set partitions.The notion of stack-sorting for set partitions was originally introduced by Defant and Kravitz.In this paper, we define a new deterministic stack-sorting map φ σ for sock sequences that use a σ-avoiding stack, where pattern containment need not be consecutive.When σ = aba, we show that our stack-sorting map sorts any sock sequence with n distinct socks in at most n iterations and that this bound is tight for n ≥ 3. We obtain a fine-grained enumeration of the number of sock patterns of length n on r distinct socks that are 1-stack-sortable under φ aba , and we also obtain asymptotics for the number of sock patterns of length n that are 1-stack-sortable under φ aba .Finally, we show that for all unsorted sock patterns σ = a • • • aba • • • a, the map φ σ cannot eventually sort all sock sequences on any multiset M unless every sock sequence on M is already sorted.
Author: Janabel Xia
Institution: Massachusetts Institute of Technology
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W4393985228