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?



Sign of the permutation which brings a subsequence back to its original form
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.


