  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 form $\mathscr{M}(\tau)$ defined by Bringmann in
her work on the Andrews $spt$-function. This formula involves a sequence of
polynomials in $j(\tau)$, through which we ultimately arrive at expressions for
the coefficients of the $j$-function purely in terms of these combinatorial
quantities.
