Algebraic relations between partition functions and the j-function

We obtain identities and relationships between the modular j-function, the generating functions for the classical partition function and the Andrews spt-function, and two functions related to unimodal sequences and a new partition statistic we call the "signed triangular weight" of a partition. These results follow from the closed formula we obtain for the Hecke action on a distinguished harmonic Maass formM(t) defined by Bringmann in her work on the Andrews spt-function. This formula involves a sequence of polynomials in j(t), through which we ultimately arrive at expressions for the coefficients of the j-function purely in terms of these combinatorial quantities.