Abstract

André 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 σ in the symmetric group S_{n} is an updown permutation if either (i) the rise set of σ consists of all the odd numbers less than n, (ii) the descent set of σ consists of all even number less than n, or (iii) both (i) and (ii). We consider analogues of André’s results for colored permutations of the form (σ, w) where σ ∈ S_{n} and w ∈ {0, …, k − 1}^{n} under the product order. That is, we define (σ_{i}, w_{i}) < (σ_{i+1}, w_{i+1}) if and only if σ_{i} < σ_{i+1} and w_{i} ≤ w_{i+1}. We then say a colored permutation (σ, w) is (I) an up-not up permutation if the rise set of (σ, w) consists of all the odd numbers less than n, (II) a not down-down permutation if the descent set of (σ, 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 shall find p, q-analogues of the generating functions for up-not up, not down-down, and up-down colored permutations.