Characters for Projective Modules in the BGG Category $mathcal{O}$ for the Orthosymplectic Lie Superalgebra $mathfrak{osp}(3|4)$

We determine the Verma multiplicities of standard filtrations of projective modules for integral atypical blocks in the BGG category $mathcal{O}$ for the orthosymplectic Lie superalgebras $mathfrak{osp}(3|4)$ by way of translation functors. We then explicitly determine the composition factor multiplicities of Verma modules using BGG reciprocity.