What's the name of the minimum number of transpositions required to build a permutation?

Question

What's the name of the minimum number of transpositions required to build a permutation? I thought it was "rank" but apparently "rank" refers to the lexicographic number.

Answer 1

I suppose it could be called length.

Answer 2

I don't think there's a fixed terminology. Here, that number is called the length of a permutation (and notice that it has a nice description in terms of the cycle-type).

I believe it is more standard for the length of a permutation to refer to the minimum number of transpositions of adjacent entries required to build a permutation. This is the same as the number of "inversions", i.e., if $\pi$ is your permutation, then the length is the number of pairs of indices $a<b$ with $\pi(a)>\pi(b)$, which is not invariant under conjugation.