Problem.Let $n\geq 2$ and let $T$ be the set of all permutations in $S_n$ of the form $$t_k:=\prod_{1\leq i\leq k/2}(i,k-i) \qquad \hbox{for $k=3,4,\ldots,n+1$}.$$ Find the least integer $f_n$ such that every $x \in S_n$ can be written as a product of at most $f_n$ elements from $T$.

This is part (ii) of Exercise 1.2.16 in Chapter 1 of *Permutation Groups* by Dixon and Mortimer (Springer 1996) and is labeled as an unsolved problem. It has been 18 years since the book was published (I guess there hasn't been any further edition yet), so I was wondering whether it is solved or not; if it is can somebody give me a hint? It looks very tough. I have solved part (i) of the exercise, showing that $T$ generates $S_n$, and showing that $2n-3$ is always an upper bound.

I asked it on math.stackexchange but did not get any answers there. I am an active member there, but here it is my first question; I hope it meets the standards here.

andMortimer. (3) The assertion you showed is exercise 1.2.16(i) there, your question is about the unsolved part (ii). $\endgroup$5more comments