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A Central Limit Theorem for Repeating Patterns

Aaron Abrams, Eric Babson, Henry Landau, Zeph Landau, James Pommersheim

Abstract

We prove a central limit theorem for the length of the longest subsequence of a random permutation which follows one of a class of repeating patterns. This class includes every fixed pattern of ups and downs having at least one of each, such as the alternating case considered by Stanley in arXiv:math/0511419 and Widom in arXiv:math/0511533. In every case considered the convergence in the limit of long permutations is to normal with mean and variance linear in the length of the permutation.

A Central Limit Theorem for Repeating Patterns

Abstract

We prove a central limit theorem for the length of the longest subsequence of a random permutation which follows one of a class of repeating patterns. This class includes every fixed pattern of ups and downs having at least one of each, such as the alternating case considered by Stanley in arXiv:math/0511419 and Widom in arXiv:math/0511533. In every case considered the convergence in the limit of long permutations is to normal with mean and variance linear in the length of the permutation.

Paper Structure

This paper contains 15 sections, 12 theorems, 19 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation
  3. Main results
  4. Patching
  5. Tracking.
  6. Patches
  7. Using patches to break the problem into pieces
  8. Proof of Main Theorem
  9. Algorithms
  10. A dynamical system
  11. Further questions
  12. Drift
  13. Computational issues
  14. Circles
  15. Appendix: Construction of $\tau$

Key Result

Theorem 4

If $w$ is combinatorial then there exist $\mu\in\mathbf R$ and $0<\sigma\in\mathbf R$ with for every $t\in \mathbf R$.

Figures (1)

  • Figure 1: The dynamical system $X$ for the pattern $UUD$.

Theorems & Definitions (28)

  • Definition 1: Patterns, length, window size. Nonconstant, simple
  • Definition 2: Following
  • Definition 3: Combinatorial, merge
  • Theorem 4
  • Corollary 5
  • Lemma 6
  • proof
  • Definition 7: Patch
  • Lemma 8
  • Lemma 9
  • ...and 18 more