In a series of papers the first author and Ono connected the rank, a
partition statistic introduced by Dyson, to weak Maass forms, a new class of
functions which are related to modular forms. Naturally it is of wide interest
to find other explicit examples of Maass forms. Here we construct a new
infinite family of such forms, arising from overpartitions. As applications we
obtain combinatorial decompositions of Ramanujan-type congruences for
overpartitions as well as the modularity of rank differences in certain
arithmetic progressions.Comment: 24 pages IMRN, accepted for publicatio