Find number of flips of transposition $(i \ j)$.

Question

Suppose that $(i \ j) \in S_n$ is a transposition and that $i < j$.

How would it be possible to find an expression for the number of flips of $(i \ j)$?

Answer 1

If I understand correctly your definition in the comments, the number of flips of an element $\sigma \in S_n$ is $nf := |\{i, j < n| i < j, x(j) < x(i)\}|$. In that case, the question seems to have an easy solution:

Suppose $(ij)$ is your transposition ($i < j$). It swaps $j$ and $i$ and leaves all other elements on their place. Now, suppose $p$ and $q$ are such numbers, that $p < q$ and $(ij)(p) > (ij)(q)$. This definitely can not happen in case, when neither $p$ nor $q$ belong to $\{i, j\}$. Neither can they in case, when $p, q \leq i$ or $p, q \geq j$. Now, suppose $p = i$ and $q \neq j$. Then $i < q < j$. Suppose $q = j$, $p \neq i$. That also results in $i < p < j$. Those cases and $p = i$, $q = j$ are the only possible ones. So $nf((ij)) = 2|\{p|i < p < j\}| + 1 = 2(|j - i| - 1) + 1 = 2|i - j| - 1$

Answer 2

The inversion number (what you call the number of flips) is the number of crossings in the arrow diagram of the permutation. This is $1$.

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