What is the smallest $k = k(n)$ for which the following holds?
There is a set of involutions $I \subset A_n$ such that for every $a \in A_n$, there exist $i_1, \ldots, i_k \in I$ such that $i_1 i_2 \cdots i_k = a$.
More generally, is there common terminology for the study of such questions? I.e., where one is not minimizing the size of the generating set, but rather minimizing the number of multiplications necessary to produce any element (of course some restriction on the generating set, e.g. only involutions, is necessary to make this question non-trivial).