Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Pin Classes I: Growth Rates

Ben Jarvis

Abstract

Pin sequences play an important role in the structural study of permutation classes. In this paper, we study the permutation classes that comprise all the finite subpermutations contained in an infinite pin sequence. We prove that these permutation classes have proper growth rates and establish a procedure for calculating these growth rates.

Pin Classes I: Growth Rates

Abstract

Pin sequences play an important role in the structural study of permutation classes. In this paper, we study the permutation classes that comprise all the finite subpermutations contained in an infinite pin sequence. We prove that these permutation classes have proper growth rates and establish a procedure for calculating these growth rates.

Paper Structure

This paper contains 28 sections, 46 theorems, 149 equations, 45 figures.

Table of Contents

  1. Basic Notions
  2. Permutations and Permutation Classes
  3. The Exponential Growth Theorem
  4. Intervals
  5. Pin Sequences
  6. Centred Permutation Classes
  7. Centred Permutations
  8. The $\boxplus$-decomposition
  9. The Generating Function Specification
  10. The Exponential Growth Theorem
  11. Examples
  12. Pin Words and Pin Classes
  13. Pin Words
  14. Pin Factors
  15. The Pin Decomposition
  16. ...and 13 more sections

Key Result

Theorem 1.7

Suppose that $\mathcal{C}$ is a proper permutation class. Then there is some constant $\mu > 0$ for which for all $n \in \mathbb{N}$. Hence $\overline{gr}(\mathcal{C})$ and $\underline{gr}(\mathcal{C})$ are both finite.

Figures (45)

  • Figure 1: The standard visual representation of the permutation $\pi = 13524$ on a $5$-by-$5$ grid. $\pi$ maps $1$ to $1$, so there is a point in the grid with coordinates $(1,1)$; $\pi$ maps $2$ to $3$, so there is a point with coordinates $(2,3)$, etc.
  • Figure 2: The above collection of points (no pair of which shares either an x- or y-coordinate) can be interpreted as representing the permutation $32451$. We don't care about the distance between points, only their relative orderings horizontally and vertically.
  • Figure 3: The permutation $213$ is contained in $13524$ (as the blue points form another copy of $213$); in turn, $13524$ is contained in $41263857$ (as the orange points form another copy of $13524$).
  • Figure 4: In the permutation $312564798$ the contiguous block $5647$ consists of four contiguous integers, so this forms an interval - as can be seen by the fact that the bounding rectangle is not sliced by any outside point in either the horizontal or vertical direction. This permutation is therefore not simple.
  • Figure 5: The permutation $3147526$, constructed from the pin word $2lurdld$. The numbers refer to the order in which the points are placed: the first point is placed in quadrant $2$ (due to the $2$ at the start of the pin word); then, the second point is placed to the left (due to the $l$) of the bounding rectangle of the first point and the origin, at the end of a pin separating point $1$ from the origin; next, point $3$ is placed above (or 'up', due ot the u) the bounding rectangle of the first two points and the origin, at the end of a pin separating point $2$ from point $1$ and the origin; and so on...
  • ...and 40 more figures

Theorems & Definitions (127)

  • Definition 1.1: Permutations
  • Definition 1.2: Permutation Containment
  • Definition 1.3: Permutation Classes
  • Definition 1.4
  • Definition 1.5: Growth Rates
  • Theorem 1.7: The Marcus-Tardos Theorem marcus:excluded-permut:
  • Theorem 1.8: Exponential Growth Theorem, Flajolet and Sedgewick flajolet:analytic-combin:
  • Theorem 1.9: Exponential Growth Theorem for Permutation Classes
  • proof
  • Definition 1.10: Intervals and Simple Permutations
  • ...and 117 more