# Combinatorics Seminar- Patterns in Random Permutations

Speaker: Chaim Even Zohar, UC Davis

Date and time: Monday, February 25, 4-5pm

Place: Rome 206

Title: Patterns in Random Permutations

Abstract: Every *k* entries in a permutation can have one of *k*! different relative orders, called patterns. How many times does each pattern occur in a large random permutation of size *n*? The distribution of this *k*!-dimensional vector of pattern densities was studied by Janson, Nakamura, and Zeilberger (2015). Their analysis showed that some component of this vector is asymptotically multinormal of order 1/sqrt(*n*), while the orthogonal component is smaller. Using representations of the symmetric group, and the theory of U-statistics, we refine the analysis of this distribution. We show that it decomposes into *k* asymptotically uncorrelated components of different orders in *n*, that correspond to representations of *S _{k}*. Some combinations of pattern densities that arise in this decomposition have interpretations as practical nonparametric statistical tests.