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\begin{document}

\begin{abstract}
Andr\'e 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 
$\sg$ in the symmetric group $S_n$ is an 
up-down permutation if either (i) the rise set of $\sg$ consists 
of all the odd numbers less than $n$, (ii) the descent set of $\sg$ 
consists of all even number less than $n$, or (iii) both (i) and (ii). 
We consider analogues of Andr\'e's results 
for colored permutations of the form $(\sg,w)$ where 
$\sg \in S_n$ and $w \in \{0,\ldots, k-1\}^n$ under the product order. 
That is, we define $(\sg_i,w_i) < (\sg_{i+1},w_{i+1})$ if and only if 
$\sg_i < \sg_{i+1}$ and $w_i \leq w_{i+1}$. 
We then say a colored permutation $(\sg,w)$ is 
(I) an {\em up-not up} permutation 
if the rise set of $(\sg,w)$ consists 
of all the odd numbers less than $n$, (II) a {\em not down-down} permutation 
if the descent set of $(\sg,w)$ consists 
of all the even numbers less than $n$, (III) an {\em up-down} permutation 
if both (I) and (II) hold. For $k \geq 2$, 
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. 
\end{abstract}


\end{document}
