Maximum size of reverse-free sets of permutations

Two words have a reverse if they have the same pair of distinct letters on the same pair of positions, but in reversed order. A set of words no two of which have a reverse is said to be reverse-free. Let F (n, k) be the maximum size of a reversefree set of words from [n]k where no letter repeats within a word. We show the following lower and upper bounds in the case n ≥ k: F (n, k) ∈ nkk−k/2+O(k/ log k). As a consequence of the lower bound, a set of n-permutations each two having a reverse has size at most nn/2+O(n/ logn).