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.