The University of Michigan Combinatorics Seminar


Abstract 

Let q=q_{1} q_{2}... q_{k} be a permutation in the symmetric group S_{k}. We say that the permutation p=p_{1} p_{2}...p_{n} in S_{n} contains a qpattern if there is a set of indices i_{q1}< i_{q2} < ... < i_{qk} such that the elements of the subsequence p_{i1}p_{i2}... p_{ik} are in the same relative order as those in q. For example, 41523 contains exactly two 132patterns, namely 152 and 153. We let
The numbers S_{n,q}(0) have been extensively studied for different patterns q and we propose looking at the sequence (S_{n,q}(c)). When q=21 this sequence counts inversions and is wellknown to be unimodal (first increasing and then decreasing). We show that this is not true in general, for any pattern in S_{3}. In fact there are an infinite number of values of n for which the sequence has internal zeros (a string of zeros with nonzero terms on either side). This is joint work with Miklós Bóna and Vincent Vatter. 