TITLE: Permutation Inversion Questions (3)
QUESTION [5 upvotes]: Working my way through a combinatorics text and I'm hung up on a couple of questions:
1.) Let $p=p_1 p_2\cdots p_n$ be a permutation.  An inversion of $p$ is a pair of entries $(p_i,p_j)$ so that $i<j$ but $p_i>p_j$.  Let us a call a permutation even (resp. odd) if it has an even (resp. odd) number of inversion.  Prove that the permuation consisting of the one cycle $(a_1a_2\cdots a_k)$ is even if $k$ is odd, and is odd if $k$ is even.
2.)  Let $I(n,k)$ be the number of $n$-permutations that have $k$ inversions.  Prove that $I(n,k)=I(n,\binom{n}{2}-k)$.
3.)  Find an explicit formula for $I(n,3)$.
These are from a combinatorics text, but I vaguely remember this topic popping up in undergrad abstract algebra and possibly in an algorithm design course as well.

REPLY [6 votes]: if the permutation consists of one cycle, (this doesn't seem to have anything to do with inversions), how many repetitions of the cycle will it take to return to identity if $k$ is even? if $k$ is odd?
for $I(n,k)$, how many total possible inversions are there? How many inversions are there of the reverse of a given permutation?
for $I(n,3)$, what is $I(n,1)$? $I(n,2)$? Look at some values, guess a formula, and use induction to quickly prove. Or think combinatorially: you have 3 inversions; if they don't interact then it's straightforward; if they do interact then a little more thought on counting them.