Sign imbalances of snakes and valley-signed permutations

One of the combinatorial structures counted by the Springer numbers is the set of snakes, which in type AnAn is the set of the alternating permutations and in type BnBn (or DnDn) is the set of certain signed permutations. The set of valley-signed permutations, defined by Josuat-Vergès, Novelli and Thibon, is another structure counted by the Springer numbers of type BnBn (or DnDn). In this paper we determine the sign imbalances of these sets of snakes and valley-signed permutations under various inversion statistics invwinvw, invoinvo, invsinvs, invBinvB, and invDinvD.