Bijections related to statistics on words

Combinatorial proofs of the identities ∑ w∈Mq^Inv(w)= n₁+n₂+⋯+n_kn₁,n₂,⋯,n₁= ∑ w∈Mq^z(w) are given and bijections are constructed between the sets {w ∈ M | Inv(w) = m}, {w ∈ M | Maj(w) = m}, {w ∈ M | Z(w) = m}, where M is the collection of all multiset permutations with n₁1's, n₂2's,...,n_kk's, Inv(w) is the inversion number of w, Maj(w) is its major index and Z(w) is the z-index of w.