Sign of the permutation which brings a subsequence back to its original form I have the following question, which I am thinking about for days now and can't get the answer right. I have a sequence of elements in this order $x_{1},x_{2},...,x_{2n}$, $n \ge 1$ and then I perform a permutation $\pi$ on this set, so it becomes $x_{\pi(1)},x_{\pi(2)},...,x_{\pi(2n)}$. Let us denote the sign of this permutation by $\varepsilon_{\pi}$.
Take the element $x_{1}$ and some arbitrary other element $x_{i}$ on the set $\{x_{2},x_{2},...,x_{2n}\}$. This is what happens now: after the permutation $\pi$, I erase the elements $x_{1}$ and $x_{i}$ of the new ordered sequence, so I end up with something like $x_{\pi(1)},x_{\pi(2)},\cdots, \hat{x}_{1},...,\hat{x}_{i},...,x_{\pi(2n)}$, where the hat $\hat{x}_{l}$ means to omit the element $x_{l}$. This new sequence has now $2n-2$ elements and I want to bring it to its original form where the subindices are ordered in increasing order, only now the elements $x_{1}$ and $x_{i}$ are obviously not present anymore. Thus I need another permutation, say $\sigma$, which permuts $2n-2$ elements, not $2n$ anymore.
So my question is: how can I relate the sign of $\sigma$ with the sign of $\pi$? Is there any relation between these two?
 A: Imagine a bubble sort where you bring each element to its original position.  $x_1$ would have taken $\pi^{-1}(1) - 1$ transpositions to bring it back to its original position.  Let's perform those transpositions, even though we'll discount them in a moment.  $x_i$ would have taken $\lvert\pi^{-1}(i) - i\rvert$ transpositions to bring it back to its original position if we had left $x_1$ in place.  Your notation $x_{\pi(1)}, \dotsc, \hat x_1, \dotsc, \hat x_i, \dotsc, x_{\pi(2n)}$ suggests you want to assume that $\pi^{-1}(i)$ is greater than $\pi^{-1}(1)$, but I assume that's just a quirk.  In any case, with $x_1$ moved back to its original position, we require $\lvert\pi^{-1}(i) - i\rvert - 1$ transpositions to move $i$ back to its proper position if $\pi^{-1}(1)$ is between $\pi^{-1}(i)$ and $i$, and $\lvert\pi^{-1}(i) - i\rvert$ otherwise.
Thus, discounting these transpositions, the sign of the permutation required to re-sort the permuted list after removing the re-placed $x_1$ and $x_i$ is $\varepsilon_\pi\cdot(-1)^{\pi^{-1}(1) + \lvert\pi^{-1}(i) - i\rvert}$ if $\pi^{-1}(1)$ is between $\pi^{-1}(i)$ and $i$, and $\varepsilon_\pi\cdot(-1)^{\pi^{-1}(1) - 1 + \lvert\pi^{-1}(i) - i\rvert}$ otherwise.
