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Permutations with a fixed number of occurrences of a pattern: A case generalizing 231

Michael Waite

Abstract

We determine a set of permutation patterns $q$ so that the number of permutations with $r$ occurrences of $q$ is asymptotically $n^r$ times the number of permutations avoiding $q$, partially settling a conjecture of Conway and Guttman. We also use these asymptotics to prove nonrationality and nonalgebraicity for certain ordinary generating functions for permutations with $r$ copies of a pattern.

Permutations with a fixed number of occurrences of a pattern: A case generalizing 231

Abstract

We determine a set of permutation patterns so that the number of permutations with occurrences of is asymptotically times the number of permutations avoiding , partially settling a conjecture of Conway and Guttman. We also use these asymptotics to prove nonrationality and nonalgebraicity for certain ordinary generating functions for permutations with copies of a pattern.
Paper Structure (6 sections, 15 theorems, 25 equations, 2 figures)

This paper contains 6 sections, 15 theorems, 25 equations, 2 figures.

Table of Contents

  1. Introduction
  2. A Lower Bound
  3. An Upper Bound
  4. Main Result
  5. Analytic Results
  6. Future directions

Key Result

Theorem 2.2

Let $q$ be a permutation of length $m$ which is indecomposable, and is of the form $q_1 \ominus ... \ominus q_k$ for some $k \geq 2$ where all of the $q_i$ are skew-indecomposable and $q_2, ..., q_{k-1}$ are not $1$. Let $q_1$ be of length $\ell$. Suppose $r \geq 1$. Then there is an injection for $n\geq rm$.

Figures (2)

  • Figure 1: Insertion of a $q$ pattern at position $s_1$, where $k = 4$
  • Figure 2: Diagram of $p_i$ and $p_j$ in $p$. The red squares contain no entries and the gray squares may contain entries.

Theorems & Definitions (24)

  • Definition 2.1
  • Theorem 2.2
  • proof
  • Corollary 2.3
  • proof
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof
  • ...and 14 more