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Permutations with a fixed number of occurrences of a monotone pattern

Michael Waite

TL;DR

This paper studies the enumeration of permutations with a fixed number $r$ of occurrences of a monotone pattern, extending the classical pattern-avoidance framework. The authors develop injection-based methods to bound $|S_{n,r}(q)|$ by a constant multiple of the corresponding avoidance count $|S_n(q)|$, first for the classical pattern $321$ and then for its generalization $321 \ominus p_0$. By composing injections with, e.g., the Simion–Schmidt map and structured decomposition, they relate $S_{n,r}(321)$ to $S_n(231)$ and, more generally, $S_{n,r}(321 \ominus p_0)$ to $S_n(231 \ominus p_0)$, yielding upper and lower bounds that are constant factors apart. Leveraging known asymptotics for monotone patterns due to Regev, they show that the generating function $\sum_{n\ge0}|S_{n,r}(k(k-1)\dots1)|z^n$ is not rational for odd $k\ge3$ and not algebraic for even $k\ge3$ (for all $r\ge1$). The methods extend prior results (e.g., Bona) and provide a unified approach to monotone-pattern enumeration with fixed pattern copies, linking pattern containment to classical avoidance classes through explicit injections and transformations.

Abstract

We bound the number of permutations with a fixed number $r$ of $321 \ominus p_0$ patterns by a constant times the number of permutations which avoid $321 \ominus p_0$. We use this new upper bound to show that the ordinary generating function for permutations with $r$ copies of $k(k-1)...1$ is not rational for odd $k \geq 3$ and not algebraic for even $k \geq 3$.

Permutations with a fixed number of occurrences of a monotone pattern

TL;DR

This paper studies the enumeration of permutations with a fixed number of occurrences of a monotone pattern, extending the classical pattern-avoidance framework. The authors develop injection-based methods to bound by a constant multiple of the corresponding avoidance count , first for the classical pattern and then for its generalization . By composing injections with, e.g., the Simion–Schmidt map and structured decomposition, they relate to and, more generally, to , yielding upper and lower bounds that are constant factors apart. Leveraging known asymptotics for monotone patterns due to Regev, they show that the generating function is not rational for odd and not algebraic for even (for all ). The methods extend prior results (e.g., Bona) and provide a unified approach to monotone-pattern enumeration with fixed pattern copies, linking pattern containment to classical avoidance classes through explicit injections and transformations.

Abstract

We bound the number of permutations with a fixed number of patterns by a constant times the number of permutations which avoid . We use this new upper bound to show that the ordinary generating function for permutations with copies of is not rational for odd and not algebraic for even .

Paper Structure

This paper contains 4 sections, 11 theorems, 35 equations, 8 figures.

Table of Contents

  1. Introduction
  2. The pattern $321$
  3. Generalizing to $321 \ominus p_0$
  4. Results for monotone patterns

Key Result

Theorem 2.5

Suppose $n,r \geq 0$. Let $b_1$ be the first entry of $q$ which takes the role of $2$ in a $321$ pattern. Suppose there are $c$ entries in $q$ to the left of $b_1$ which are greater than $b_1$, and $a$ entries in $q$ to the right of $b_1$ which are smaller than $b_1$. Then there exist $0 \leq s,t <

Figures (8)

  • Figure 1: An example of $p = w_1\beta_1 w_2$ with $\alpha_i$ and $\gamma_i$ entries labeled
  • Figure 2: An example of $\sigma$ and $\tau$
  • Figure 3: An example of $\text{reduce}(\sigma)$ and $\text{reduce}(\sigma)$.
  • Figure 4: An example of $\text{reduce}(\sigma)$ and $\text{reduce}(\tau)$
  • Figure 5: An example of $\varphi^q(p)$
  • ...and 3 more figures

Theorems & Definitions (24)

  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Example 2.4
  • Theorem 2.5
  • proof
  • Theorem 2.6
  • proof
  • Definition 3.1
  • Example 3.2
  • ...and 14 more