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Arrow pattern avoidance in permutations: structure and enumeration

Kassie Archer, Robert P. Laudone

Abstract

Arrow patterns were introduced by Berman and Tenner as a generalization of vincular patterns. They observed that arrow patterns have the potential to bridge the divide between a permutation's cycle notation and its one-line notation; in support of this, they used arrow avoidance to enumerate shallow and cyclic shallow permutations. More recently, $321$-avoiding cyclic permutations were recharacterized entirely in terms of arrow avoidance. Motivated by these results, we initiate a systematic study of arrow avoidance. In this paper, we prove structural results about arrow patterns, including defining arrow-Wilf equivalence, and enumerate several arrow avoidance classes. Finally, we consider the avoidance of pairs of arrow patterns, focusing on cases that prohibit fixed points in the underlying permutation.

Arrow pattern avoidance in permutations: structure and enumeration

Abstract

Arrow patterns were introduced by Berman and Tenner as a generalization of vincular patterns. They observed that arrow patterns have the potential to bridge the divide between a permutation's cycle notation and its one-line notation; in support of this, they used arrow avoidance to enumerate shallow and cyclic shallow permutations. More recently, -avoiding cyclic permutations were recharacterized entirely in terms of arrow avoidance. Motivated by these results, we initiate a systematic study of arrow avoidance. In this paper, we prove structural results about arrow patterns, including defining arrow-Wilf equivalence, and enumerate several arrow avoidance classes. Finally, we consider the avoidance of pairs of arrow patterns, focusing on cases that prohibit fixed points in the underlying permutation.
Paper Structure (8 sections, 32 theorems, 31 equations, 6 tables)

This paper contains 8 sections, 32 theorems, 31 equations, 6 tables.

Table of Contents

  1. Introduction
  2. Definitions and notation
  3. Arrow patterns
  4. Permutations that avoid $(12;b\to c)$
  5. Permutations that avoid $(21;b\to c)$
  6. Permutations that avoid $(13;b\to c)$
  7. Avoiding pairs of arrow patterns
  8. Conjectures and Open Questions

Key Result

Lemma 1.1

For $n \geq 1$,

Theorems & Definitions (73)

  • Lemma 1.1: Cl01
  • Proposition 2.1
  • proof
  • Theorem 2.2
  • proof
  • Proposition 2.3
  • proof
  • Theorem 2.4
  • proof
  • Lemma 2.5
  • ...and 63 more