  The symmetric $\mathfrak{gl}_n$-homologies, introduced by Robert and Wagner,
provide a categorification of the Reshetikhin--Turaev invariants corresponding
to symmetric powers of the standard representation of quantum
$\mathfrak{gl}_n$. Unlike in the exterior setting, these homologies are already
non-trivial when $n=1$. Moreover, in this case, their construction can be
greatly simplified. Our first aim is giving a down-to-earth description of the
non-equivariant symmetric $\mathfrak{gl}_1$-homology, together with relations
that hold in this setting. We then find a basis for the state spaces of graphs,
and use it to construct an algorithm and a program computing the invariant for
uncolored links.
