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 (2008) and Widom (2006). 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.