Express reverse inversion, major index, descents in terms of the forward direction. Given $w=a_1a_2...a_n \in S_n $, then the reverse of $w$ is $w^r=a_n....a_2a_1$.  
Express inv($w^r$), des($w^r$) and maj($w^r$) in terms inv($w$), des($w$), maj($w$), respectively.    
I know the definition of an inversion is a partition(i, j) s.t. i $<$ j and part(i) > part(j), descent is where the number goes down, and major index is  the sum of the positions of the descents of the permutation.  
Can someone point me in the right direction as to how to go about solving these? 
Perhaps some formulas, or theorems.

HINT: For the inversions, show that $\langle i,j\rangle$ is an inversion of $w^r$ if and only if it is not an inversion of $w$. How many pairs $\langle i,j\rangle$ with $i<j$ are there?
For the descents, show that each descent of $w^r$ is an ascent of $w$ and vice versa, so $w^r$ has exactly as many descents as $w$ has ascents. If $w$ has $k$ descents, how many ascents does it have?
For the major index, show that if $i$ is an ascent of $w$, then the corresponding descent of $w^r$ is $n-i$. Thus, if $s$ is the sum of the ascents of $w$, and $w$ has $a$ ascents, then $\operatorname{maj}(w^r)=an-s$. You know $a$ from the second part, so it only remains to find $s$ in terms of $n$ and $\operatorname{maj}(w)$. Further HINT: Express $\operatorname{maj}(w)+s$ in terms of $n$.
