In this note we introduce a new class of refined Eulerian polynomials defined by
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<!-- MATH: \begin{displaymath}
A_n(p,q)=\sum_{\pi\in\mathfrak{S} _{n}}p^{{{\rm odes}\,}(\pi)}q^{{{\rm edes}\,}(\pi)},
\end{displaymath} -->


<IMG
 WIDTH="244" HEIGHT="65"
 SRC="img2.gif"
 ALT="\begin{displaymath}A_n(p,q)=\sum_{\pi\in\mathfrak{S} _{n}}p^{{{\rm odes}\,}(\pi)}q^{{{\rm edes}\,}(\pi)},\end{displaymath}">
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where 
<!-- MATH: ${{\rm odes}\,}(\pi)$ -->
<IMG
 WIDTH="69" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="img3.gif"
 ALT="${{\rm odes}\,}(\pi)$">
and 
<!-- MATH: ${{\rm edes}\,}(\pi)$ -->
<IMG
 WIDTH="67" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="img4.gif"
 ALT="${{\rm edes}\,}(\pi)$">
enumerate the number of descents of permutation <IMG
 WIDTH="15" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="img5.gif"
 ALT="$\pi$">
in odd and even positions, respectively.
We show that the refined Eulerian polynomials 
<!-- MATH: $A_{2k+1}(p,q),k=0,1,2,\ldots,$ -->
<IMG
 WIDTH="219" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="img6.gif"
 ALT="$A_{2k+1}(p,q),k=0,1,2,\ldots,$">
and 
<!-- MATH: $(1+q)A_{2k}(p,q),k=1,2,\ldots,$ -->
<IMG
 WIDTH="240" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="img7.gif"
 ALT="$(1+q)A_{2k}(p,q),k=1,2,\ldots,$">
have a nice symmetry property.