Bringing a permutation back to the identity I'm working with transposition distance (nothing to do with algebraic transpositions) on given permutations. Given a permutation, how many moves (transpositions) will it take to get back to the identity. A transposition exchanges consecutive strings in a permutation. For example, consider the following permutation:
$$(\begin{matrix}\underline{3} & \underline{2 \ \ \ \ 1} & 5 & 4\end{matrix}) \rightarrow
(\begin{matrix} \underline{2} & \underline{1} & 3 & 5 & 4\end{matrix}) \rightarrow
(\begin{matrix} 1 & 2 & 3 & \underline{5} & \underline{4}\end{matrix}) \rightarrow
(\begin{matrix} 1 & 2 & 3 & 4 & 5\end{matrix}).$$
It takes a minimum of 3 moves to return it to the identity, so we say it has transposition distance 3. Now, what is the transposition distance of the following permutation:
$$(\begin{matrix} 5 & 4 & 3 & 2 & 1\end{matrix})?$$
I say it has transposition distance 4, but my book (Combinatorics of Genome Rearrangements) says that no permutations of length 5 have t.d. 4. Can anyone find a way to bring that permutation back to the identity in no more than 3 moves?

Halfway through a meet-in-the-middle brute force search, I found:
$$ \underline{5~4}~\underline{3~2}~1 \to 3~\underline{2~5}~\underline{4~1}
\to \underline{3~4}~\underline{1~2}~5 \to 1~2~3~4~5 $$
