Sequence of permutations without a fixed point

Question

I need to find $m$ permutations $A_1,..,A_m$ where each $A_i$ is a permutation on $n$ objects such that any of the compositions $A_jA_{j+1}..A_{j+k}$; $1 \leq j \leq n-1$ and $j+k \leq m$ does not have a fixed point.

Does such a sequence of permutation always exist? If yes, is there a good way to generate such permutations? Do we have a name for such a sequence of permutations? Thanks for your time.

Answer 1

Of course such permutations exist if $n > m$ (just take each $A_i$ to be the same $n$-cycle). On the other hand if $m \le n$, a pigeonhole argument should give some pair of $A_1, A_1 A_2, A_1A_2A_3, \ldots$ mapping element $1$ to the same element, inducing a fixed point in the quotient of this pair. Are you sure this question is well-posed?