Question

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 $\theta$?

Answer 1

It seems like there is no hope for a nice closed form for $F(q,\theta)=\sum_{\sigma\in S_n}q^{l(\sigma)}\theta^{[\sigma]}$. When the sum is restricted to $\sigma\in S_n$ which are involutions, the computation can be found in I. Gessel's paper "A q-analog of the exponential formula".

For the general case, the study of $F(q,\theta)$ is the main topic of P.H. Edelman's paper "On inversions and cycles in permutations" (Europ. J. Combinatorics,(1987) vol 8, 269-279). he proves a bunch of properties of this bivariate generating function, yet, according to this paper even computing the number of permutations which achieve the minimum number of cycles with a fixed number of inversions hasn't been carried. It gives a bunch of open problems about $F(q,\theta)$.

If on the other hand you let $l(\sigma)$ denote the number of inversions of $\sigma$ written in cycle notation (i.e. the number of inversions in $(a_1\dots a_{k_1})(a_{k_1+1}\dots a_{k_2})\cdots(a_{k_{r}+1}\dots k_{r+1})$) then the sum is $\prod _{i=1}^n [i] _{q} ^{\theta}$, where $$[i]_{q}^{\theta}= 1+q+\cdots+q^{i-2}+\theta q^{i-1}.$$ This is proved in "Cycles and patterns in permutations" by R. Parviainen.