A Central Limit Theorem for Repeating Patterns

TL;DR

This paper establishes a central limit theorem for the length of the longest subsequence following certain repeating patterns in random permutations, showing normal convergence with linear mean and variance.

Contribution

It extends the CLT to a broad class of repeating patterns in permutations, including alternating patterns, with explicit asymptotic normality results.

Findings

Longest pattern subsequence length converges to normal distribution.

Mean and variance grow linearly with permutation length.

Applicable to various repeating pattern classes, including alternating 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.