  We compute the exact density of states and 2-point function of the
$\mathcal{N} =2$ super-symmetric SYK model in the large $N$ double-scaled
limit, by using combinatorial tools that relate the moments of the distribution
to sums over oriented chord diagrams. In particular we show how SUSY is
realized on the (highly degenerate) Hilbert space of chords. We further
calculate analytically the number of ground states of the model in each charge
sector at finite $N$, and compare it to the results from the double-scaled
limit. Our results reduce to the super-Schwarzian action in the low energy
short interaction length limit. They imply that the conformal ansatz of the
2-point function is inconsistent due to the large number of ground states, and
we show how to add this contribution. We also discuss the relation of the model
to $SL_q(2|1)$. For completeness we present an overview of the $\mathcal{N}=1$
super-symmetric SYK model in the large $N$ double-scaled limit.
