Above is just an example I'm trying to work from as I have the solutions. I've seen lots of definitions of what inversions are but they use signs like sigma, and it doesn't really explain what the sigma is. So if anyone could just give me a easy to understand definition and use the above as a possible example that would be great. Thank you.
In my book it says it says it has 3 inversions (3,1),(3,2),(2,1), how would these be found?
On
Let me quote from the Wikipedia:
Formally, let $(A(1), \ldots, A(n))$ be a sequence of $n$ distinct numbers. If $i < j$ and $A(i) > A(j)$, then the pair $(i, j)$ is called an inversion of $A$.
(It seems in this case you write that $(j, i)$ is an inversion, but this is not an important difference.)
So you should check for all pairs $(i, j)$, with $1 \le i < j \le 5$ whether $A(i) > A(j)$, where $A(i)$ denotes the number under $i$ in the representation of your permutation.
An inversion of a permutation $f$ (of $\{1,2,\dots,n\}$) is a pair of elements $(i,j)$ whose relative order gets reversed when you apply $f$. Different authors use different conventions about how to make this precise, and your example seems to be compatible with two of these conventions.
Convention 1: An inversion is a pair $(i,j)$ such that $i>j$ but $f(i)<f(j)$.
Convention 2: An inversion is a pair $(f(i),f(j))$ where $i<j$ but $f(i)>f(j)$.
(The inversions of $f$ in one of these conventions would be the inversions of $f^{-1}$ in the other convention. In your example, $f^{-1}=f$, so there's not enough information to tell which convention is being used.)
Another convention that seems to be pretty widely used is:
Convention 3: An inversion is a pair $(i,j)$ where $i<j$ but $f(i)>f(j)$.
By the way, all these conventions agree as to the number of inversions of $f$.