Two Permutation Classes related to the Bubble Sort Operator

We introduce the Dual Bubble Sort operator B̂ (a sorting algorithm such that, if σ = α 1β is a permutation, then B̂(σ) = 1α B̂(β)) and consider the set of permutations sorted by the composition B̂B, 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 B̂2 and B2.