Rational homological stability for groups of partially symmetric automorphisms of free groups

Let F_{n+m} be the free group of rank n+m, with generators x_1,...,x_{n+m}. An automorphism \phi of F_{n+m} is called partially symmetric if for each 1 \le i \le m, \phi(x_i) is conjugate to x_j or x_j^{-1} for some 1 \le j \le m. Let \Sigma\Aut_n^m be the group of partially symmetric automorphisms. We prove that for any m \ge 0 the inclusion \Sigma\Aut_n^m \to \Sigma\Aut_{n+1}^m induces an isomorphism in rational homology for dimensions i satisfying n \ge (3(i+1)+m)/2, with a similar statement for the groups P\Sigma\Aut_n^m of pure partially symmetric automorphisms. We also prove that for any n \ge 0 the inclusion \Sigma\Aut_n^m \to \Sigma\Aut_n^{m+1} induces an isomorphism in rational homology for dimensions i satisfying m > (3i-1)/2.