I am interested in the orders of products of $(n-1)$-cycles in the symmetric group $S_n$. In particular what orders of elements can occur as products of $m$ $(n-1)$-cycles for $m=2, 3, 4,\ldots $? It would be even better to know which cycle structures products of $m$ $(n-1)$-cycles in $S_n$ can have for $m=2, 3, 4,\ldots $.
I would like it to be the case that if $n=2^k + 1$, one must multiply $2^{k-1}$ or more $2^k$-cycles for there to be an element of order 2 expressed as a product of $2^k$-cycles.
There certainly are element of order 2 expressed in this way: simply raise a $2^k$ cycle to the $2^{k-1}$ power and the result is of order 2. What I hope is that one cannot get an element of order 2 as a product of fewer than $2^{k-1}$ $2^k$-cycles by allowing different $2^k$ cycles in the product.
Even a method for attacking general the problem or the specific case I'm most interested in would be helpful.
The problem arose in connection with a construction I am trying to do in quandle theory.