# Consecutive Up-Down Patterns in Up-Down Permutations

Keywords:
up-down permutations, consecutive patterns, generating functions

### Abstract

In this paper, we study the distribution of the number of consecutive pattern matches of the five up-down permutations of length four, $1324$, $2314$, $2413$, $1432$, and $3412$, in the set of up-down permutations. We show that for any such $\tau$, the generating function for the distribution of the number of consecutive pattern matches of $\tau$ in the set of up-down permutations can be expressed in terms of what we call the generalized maximum packing polynomials of $\tau$. We then provide some systematic methods to compute the generalized maximum packing polynomials for such $\tau$.