Deterministic stack-sorting for set partitions
Janabel Xia

TL;DR
This paper introduces a new deterministic stack-sorting method for sock sequences, characterizes its sorting bounds, and enumerates sortable sock patterns, advancing understanding of pattern-avoiding stack-sorting for set partitions.
Contribution
It defines a novel $\sigma$-avoiding stack-sorting map for sock sequences and analyzes its sorting efficiency and pattern enumeration, especially for $\sigma=aba$.
Findings
The map sorts any sock sequence with n distinct socks in at most n iterations.
The number of 1-stack-sortable sock patterns under $\sigma=aba$ is explicitly enumerated and asymptotically analyzed.
For most patterns, the sorting map cannot sort all sequences unless they are already sorted.
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 for sock sequences that uses a -avoiding stack, where pattern containment need not be consecutive. When , we show that our stack-sorting map sorts any sock sequence with distinct socks in at most iterations, and that this bound is tight for . We obtain a fine-grained enumeration of the number of sock patterns of length on distinct socks that are -stack-sortable under…
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Taxonomy
Topicssemigroups and automata theory · Algorithms and Data Compression · Advanced Combinatorial Mathematics
