Two permutation classes related to the Bubble Sort operator

We introduce the Dual Bubble Sort operator (B) over capB (a sorting algorithm such that, if sigma = alpha 1 beta is a permutation, then (B) over cap(sigma) = 1 alpha(B) over cap(beta)) and consider the set of permutations sorted by the composition (B) over cap B, where B is the classical Bubble Sort operator. We show that this set is a permutation class and we determine the generating function of the descent and fixed point distributions over this class. Afterwards, we characterize the same distributions over the set of permutations that are sorted by both (B) over cap (2) and B-2.