  We consider the pseudo-Anosov elements of the mapping class group of a
surface of genus g that fix a rank k subgroup of the first homology of the
surface. We show that the smallest entropy among these is comparable to
(k+1)/g. This interpolates between results of Penner and of Farb and the second
and third authors, who treated the cases of k=0 and k=2g, respectively, and
answers a question of Ellenberg. We also show that the number of conjugacy
classes of pseudo-Anosov mapping classes as above grows (as a function of g)
like a polynomial of degree k.
