Analogues of Up-down Permutations for Colored Permutations

Andre proved that sec x is the generating function of all up-down permutations of even length and tan x is the generating function of all up-down permutation of odd length. There are three equivalent ways to define up-down permutations in the symmetric group S-n. That is, a permutation sigma in the symmetric group S-n is an up-down permutation if either (i) the rise set of sigma consists of all the odd numbers less than n, (ii) the descent set of sigma consists of all even number less than n, or (iii) both (i) and (ii). We consider analogues of Andre's results for colored permutations of the form (sigma,w) where sigma epsilon S-n and w epsilon {0, . . . , k-1}(n) under the product order. That is, we define (sigma i , w(i)) < (sigma(i+1) ,w(i+1)) if and only if sigma(i) < sigma(i+1) and w(i) <= w(i+1). We then say a colored permutation (sigma,w)is (I) an up-not up permutation if the rise set of (sigma,w) consists of all the odd numbers less than n, (II) a not down-down permutation if the descent set of (sigma,w) consists of all the even numbers less than n, (III) an up-down permutation if both (I) and (II) hold. For k >= 2, conditions (I), (II), and (III) are pairwise distinct. We find p, q-analogues of the generating functions for up-not up, not down-down, and up-down colored permutations.