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Stack-sorting with Stacks Avoiding Vincular Patterns

William Zhao

Abstract

We introduce the stack-sorting map $\text{SC}_σ$ that sorts, in a right-greedy manner, an input permutation through a stack that avoids some vincular pattern $σ$. The stack-sorting maps of Cerbai et al. in which the stack avoids a pattern classically and Defant and Zheng in which the stack avoids a pattern consecutively follow as special cases. We first characterize and enumerate the sorting class $\text{Sort}(\text{SC}_σ)$, the set of permutations sorted by $s\circ\text{SC}_σ$, for seven length $3$ patterns $σ$. We also decide when $\text{Sort}(\text{SC}_σ)$ is a permutation class. Next, we compute $\max_{π\in \mathfrak S_n}|\text{SC}_σ^{-1}(π)|$ and characterize the periodic points of $\text{SC}_σ$ for several length $3$ patterns $σ$. We end with several conjectures and open problems.

Stack-sorting with Stacks Avoiding Vincular Patterns

Abstract

We introduce the stack-sorting map that sorts, in a right-greedy manner, an input permutation through a stack that avoids some vincular pattern . The stack-sorting maps of Cerbai et al. in which the stack avoids a pattern classically and Defant and Zheng in which the stack avoids a pattern consecutively follow as special cases. We first characterize and enumerate the sorting class , the set of permutations sorted by , for seven length patterns . We also decide when is a permutation class. Next, we compute and characterize the periodic points of for several length patterns . We end with several conjectures and open problems.

Paper Structure

This paper contains 17 sections, 65 theorems, 39 equations, 5 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definitions and Notation
  4. Preliminary Lemmas
  5. The sorting classes $\text{Sort}(\sc_\sigma)$
  6. The Sorting Classes of $\sc_{1\underline{23}}$, $\sc_{\underline{32}1}$, and $\sc_{3\underline{21}}$
  7. The Sorting Classes of $\sc_{\underline{13}2}$ and $\sc_{\underline{23}1}$
  8. The Sorting Classes of $\sc_{1\underline{32}}$ and $\sc_{\underline{12}3}$
  9. The remaining maps $\sc_\sigma$
  10. Dynamics of the maps $\sc_\sigma$
  11. Maximum preimages of the maps $\sc_\sigma$
  12. The maps $\sc_{\underline{12}3}$ and $\sc_{\underline{32}1}$
  13. The maps $\sc_{\underline{23}1}$ and $\sc_{\underline{21}3}$
  14. The maps $\sc_{2\underline{31}}$ and $\sc_{2\underline{13}}$
  15. The maps $\sc_{1\underline{23}}$ and $\sc_{3\underline{21}}$
  16. ...and 2 more sections

Key Result

Lemma 2.1

For a pattern $\sigma\in \mathcal{S}$ of length at least $2$, we have

Figures (5)

  • Figure 1: The stack-sorting map $s_{123}=\sc_{123}$ acting on $\tau=514362$
  • Figure 2: The stack-sorting map $\sc_{\underline{123}}$ acting on $\tau=514362$
  • Figure 3: The stack-sorting map $\sc_{\underline{12}3}$ acting on $\tau=514362$
  • Figure 4: The stack-sorting map $\sc_{1\underline{23}}$ acting on $\tau=514362$
  • Figure 5: The mesh patterns $\mu_{132}$ and $\mu_{2413}$.

Theorems & Definitions (97)

  • Lemma 2.1: Defant
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • Lemma 3.1: Knuth
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof
  • Proposition 3.4
  • ...and 87 more