  We reconsider an old problem, namely the dimension of the $G$-invariant
subspace in $V^{\otimes p} \otimes V^{*\otimes q}$, where $G$ is one of the
classical groups ${\rm GL}(V)$, ${\rm SL}(V)$, ${\rm O}(V)$, ${\rm SO}(V)$, or
${\rm Sp}(V)$. Spanning sets for the invariant subspace have long been well
known, but linear bases are more delicate. Beginning in the broader setting of
polynomial invariants, we write down multigraded linear bases which we realize
as certain arc diagrams (which may include hyperedges); then the bases for
tensor invariants are obtained by restricting our attention to the diagrams
that are 1-regular. We survey the many equivalent ways -- some old, some new --
to enumerate these diagrams.
