I am looking for an answer to the following problem.
Let $S$ be the multiset $\{1^{d_1},2^{d_2},\dots,m^{d_m}\}$ $A_{S,k}$ is the number of permutations of $S$ with $k-1$ descents and no descent at the end.
Let $A_S(t)=\sum\limits_k A_{S,k}t^k$ and $d=\sum\limits_{i\leq m} d_i$. I want to show that $$\sum\limits_S \frac{A_S(t)}{(1-t)^{d+1}} \prod\limits_i x_i^{d_i}=(1-t\prod\limits_i(i-x_i)^{-1})^{-1}$$
The Eulerian polynomial $A_d(t)$ is the special case $A_{[d]}$. I want to show that $$\sum\limits_{d=0}^{\infty}\frac{A_d(t)x^d}{(1-t)^{d+1}d!}=\frac{1}{1-te^x}$$
Thanks for your help.