A Foata bijection for the alternating group and for q analogues
  The Foata bijection $\Phi : S_n \to S_n$ is extended to the bijections $\Psi : A_{n+1} \to A_{n+1}$ and $\Psi_q : S_{n+q-1} \to S_{n+q-1}$, where S_m, A_m are the symmetric and the alternating groups. These bijections imply bijective proofs for recent equidistribution theorems, by Regev and Roichman, for A_{n+1} and for S_{n+q-1}. 
Dan Bernstein, Amitai Regev