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.



Sequence of permutations without a fixed point
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?


