  The universal enveloping algebra of ${\cal W}_{1+\infty}$ is isomorphic to
the affine Yangian of $\mathfrak{gl}_1$. We study the ${\cal N}=2$
supersymmetric version of this correspondence, and identify the full set of
defining relations of the supersymmetric affine Yangian. These relations can be
deduced by demanding that the algebra has a representation on
twin-plane-partitions, which we construct by gluing pairs of plane partitions.
We define the action of the algebra on these twin-plane-partitions explicitly.
