Generation of iterated wreath products constructed from alternating, symmetric and cyclic groups

Let [Formula: see text], [Formula: see text] be a sequence of groups each of which is either an alternating group, a symmetric group or a cyclic group. Let us construct a sequence [Formula: see text] of wreath products via [Formula: see text] and, for each [Formula: see text], [Formula: see text] via the natural permutation action. We determine the minimum number [Formula: see text] of generators required for each wreath product in this sequence.