This question is inspired but not directly related to this recent [Stanley's MO post](https://mathoverflow.net/questions/419698/number-of-sets-s-for-which-number-of-permutations-in-s-n-with-descent-set-s).


The *descent set* $D(w)$ of a permutation $w=a\_1 a\_2\cdots a\_n\in\frak{S}\_n$ (the symmetric group on $\{1,\dots,n\}$) is
defined by $D(w)=\{ 1\leq i\leq n-1\,:\, a\_i>a\_{i+1}\}$. Denote the cardinality of $D(w)$ by $des(w)$.


The [Eulerian polynomials](https://en.wikipedia.org/wiki/Eulerian_number) are known to express
$$A\_n(x)=\sum\_{w\in\frak{S}\_n}x^{des(w)}$$
whose exponential generating function is given by
$$\sum\_{n\geq0}A\_n(x)\,\frac{z^n}{n!}=\frac{1-z}{z-e^{(1-z)x}}.$$
Suppose we wish to refine this by including the *cycle-type* of each permutation.



> 
> **QUESTION.** Is there a generating function for the sum
> $$\sum\_{w\in\frak{S}\_n}x^{des(w)}\,t\_1^{a\_1}t\_2^{a\_2}\cdots t\_n^{a\_n}$$
> where $(a\_1,\dots,a\_n)$ is the cycle-type of $w$? **Caveat.** the $a\_i$'s are non-negative integers.
> 
> 
> 


**Examples.** The first few polynomials for $n=1, 2, 3$ and $4$:
\begin{align\*} &t\_1 \\ &t\_1^2+t\_2x \\ &t\_1^3+(2t\_1t\_2+2t\_3)x+t\_1t\_2x^2 \\
&t\_1^4+(3t\_1^2t\_2+4t\_1t\_3+t\_2^2+3t\_4)x+(3t\_1^2t\_2+4t\_1t\_3+t\_2^2+3t\_4)x^2+t\_2^2x^3.
\end{align\*}



Generating function for "descents" and "cycle-types", in tandem
Counting permutations by cycle type and descents was first accomplished in I. Gessel and C. Reutenauer, *Counting permutations with given cycle structure and
descent set*, J. Combin. Theory Ser. A 64, No. 2 (1993), 189–215.


Using the results of this paper, an explicit (though somewhat complicated) formula for the generating function asked for can be found in
Jason Fulman, *The distribution of descents in fixed conjugacy classes of the symmetric groups*, J. Combin. Theory Ser. A 84 (1998), no. 2, 171–180. Another proof of Fulman's formula is Theorem 7.1 (c) in
Ira M. Gessel and Yan Zhuang,
*Plethystic formulas for permutation enumeration,*
Adv. Math. 375 (2020), 107370, 55 pp.


