Singular invariants and coefficients of weak harmonic Maass forms of weight 5/2

We study the coefficients of a natural basis for the space of weak harmonic Maass forms of weight $5/2$ on the full modular group. The non-holomorphic part of the first element of this infinite basis encodes the values of the partition function $p(n)$. We show that the coefficients of these harmonic Maass forms are given by traces of singular invariants. These are values of non-holomorphic modular functions at CM points or their real quadratic analogues: cycle integrals of such functions along geodesics on the modular curve. The real quadratic case relates to recent work of Duke, Imamo\=glu, and T\'oth on cycle integrals of the $j$-function, while the imaginary quadratic case recovers the algebraic formula of Bruinier and Ono for the partition function.