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{\bf John Hall, Jeffrey Liese and Jeffrey B. Remmel}
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{\bf $q$-Counting Descent Pairs with Prescribed Tops and Bottoms}
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Given sets $X$ and $Y$ of positive integers and a permutation
$\sigma = \sigma_1 \sigma_2 \cdots \sigma_n \in S_n$, an
$(X,Y)$-\emph{descent} of $\sigma$ is a descent pair $\sigma_i >
\sigma_{i+1}$ whose ``top'' $\sigma_i$ is in $X$ and whose
``bottom'' $\sigma_{i+1}$ is in $Y$. Recently Hall and Remmel
proved two formulas for the number $P_{n,s}^{X,Y}$
of $\sigma \in S_n$ with $s$ $(X,Y)$-descents, which generalized
Liese's results in [1].  We define a new statistic
${\rm stat}_{X,Y}(\sigma)$ on permutations $\sigma$ and define
$P_{n,s}^{X,Y}(q)$ to be the sum of $q^{{\rm stat}_{X,Y}(\sigma)}$
over all $\sigma \in S_n$ with $s$ $(X,Y)$-descents. We then show
that there are natural $q$-analogues of the Hall-Remmel formulas for
$P_{n,s}^{X,Y}(q)$.



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