Consecutive up-down patterns in up-down permutations (Q405276)

Summary: 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 \(\tau\).