Suppose it is possible, through many steps, to move from the permutation $\pi$ to the permutation $\sigma$ by multiplying, at each step, by a transposition.
Knowing that they are not necessarily conjugates, I want to find:
In how many ways (The maximum number) I can move from $\pi$ to $\sigma$.
bound the maximum number of steps needed to switch from $\pi$ to $\sigma$ (is there a reference).
EDIT 2 In other words, as Peter Taylor proposed:
Let $d(\pi,\sigma)$ be the length of the smallest sequence of transpositions we need to multiply by the permutation $\pi$ to take it to $\sigma$.
How can I bound or even calculate the number of sequences of length $d(\pi,\sigma)$ which take $\pi$ to $\sigma$? (I am looking for any idea)
How can I bound or even calculate $d(\pi,\sigma)$ (This one is answered as explained above)?
It is known that the number of transpositions in the symmetric group $S_n$ is $\frac{n(n-1)}{2}$ and also it is possible to decompose a certain permutation into disjoint cycles, but I don't how to collect these results to answer my question!
Any kind of help is appreciated.
EDIT 1: I don't think my question is a duplicate! it consists of two parts and the marked question answers only one question! The linked question is not talking about the first part of my question!