What is the value of sign($\tau$)?

Question

Let $\tau\in S(p+q)$ be the permutation with

$\tau(1)=p+1,\;\tau(2)=p+2,\;\ldots\;,\;\tau(q)=p+q$

$\tau(q+1)=1,\;\tau(q+2)=2,\;\ldots\;,\;\tau(q+p)=p$

How can I calculate the sign of this permutation?

Thanks!

Answer 1

You can determine the number of inversions, that is the number of pairs $(i,j)$ with $i<j$ and $\tau(i)>\tau(j)$. These inversions are the pairs with $1\le i\le q$ and $q+1\le j\le p+q$, so there are $pq$ of them, and the sign of $\tau$ is $(-1)^{pq}$.