By the Bruhat decomposition of $GL(n, \mathbb{F}_q) / B_n$ we know that $$[n]! = \sum_{ \sigma \in S(n)} q^{l(\sigma)}$$ where $[n]! = \prod_{j=1}^n (1+q + \cdots + q^{j-1})$ and $l(\sigma)$ is the length of the permutation $\sigma \in S(n)$ (also known as the number of involutions of $\sigma$).

We also know that $$\theta^{(n)} = \sum_{\sigma \in S(n)} \theta^{[\sigma]}$$ where $[\sigma]$ is the number of cycles of the permutation $\sigma \in S(n)$ and $\theta^{(n)} = \theta(\theta+1) \cdots (\theta+n-1)$. Notice that $$\lim_{q \rightarrow 1} \ [n]! = n! = \lim_{\theta \rightarrow 1} \ \theta^{(n)}.$$ Is there a way to write $$\sum_{\sigma \in S(n)} q^{l (\sigma)} \theta^{[\sigma]}$$ explicitly as a function of $q$ and $\theta$?