Consecutive Up-down Patterns in Up-down Permutations

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 τ , the generating function for the distribution of the number of consecutive pattern matches of τ in the set of up-down permutations can be expressed in terms of what we call the generalized maximum packing polynomials of τ . We then provide some systematic methods to compute the generalized maximum packing polynomials for such τ .