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On the set partitions that require maximum sorts through the $aba-$avoiding stack

Yunseo Choi, Katelyn Gan, Andrew Li, Tiffany Zhu

Abstract

Recently, Xia introduced a deterministic variation $φ_σ$ of Defant and Kravitz's stack-sorting maps for set partitions and showed that any set partition $p$ is sorted by $φ^{N(p)}_{aba}$, where $N(p)$ is the number of distinct alphabets in $p$. Xia then asked which set partitions $p$ are not sorted by $φ_{aba}^{N(p)-1}$. In this note, we prove that the minimal length of a set partition $p$ that is not sorted by $φ_{aba}^{N(p)-1}$ is $2N(p)$. Then we show that there is only one set partition of length $2N(p)$ and ${{N(p) + 1} \choose 2} + 2{N(p) \choose 2}$ set partitions of length $2N(p)+1$ that are not sorted by $φ_{aba}^{N(p)-1}$.

On the set partitions that require maximum sorts through the $aba-$avoiding stack

Abstract

Recently, Xia introduced a deterministic variation of Defant and Kravitz's stack-sorting maps for set partitions and showed that any set partition is sorted by , where is the number of distinct alphabets in . Xia then asked which set partitions are not sorted by . In this note, we prove that the minimal length of a set partition that is not sorted by is . Then we show that there is only one set partition of length and set partitions of length that are not sorted by .
Paper Structure (3 sections, 11 theorems, 7 equations, 2 figures)

This paper contains 3 sections, 11 theorems, 7 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proofs of the Main Results

Key Result

Theorem 1.1

If set partition $p$ satisfies $|p| \leq 2N(p)$ for some $N(p) \geq 3$ and is not sorted after applying $\phi^{N(p)-1}_{aba}$, then $p$ is equivalent to $(a_1 a_2 \cdots a_{N(p)})^2.$

Figures (2)

  • Figure 1: West's stack-sorting map $s$ on $\pi = 4213$
  • Figure 2: Xia's stack-sorting map $\phi_{aba}$ on $p= abcac$

Theorems & Definitions (18)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1: Xia xia2023deterministic
  • Corollary 2.2: Xia xia2023deterministic
  • Corollary 2.3
  • Proposition 3.1
  • proof : Proof of \ref{['main']}
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 8 more