  In this paper, we systematically study the measures of multi-partite
entanglement with the aim of constructing those measures that can be computed
in probe approximation in the holographic dual. We classify and count general
measures as invariants of local unitary transformations. After formulating
these measures in terms of permutation group elements, we derive conditions
that a probe measure should satisfy and find a large class of solutions. These
solutions are generalizations of the multi-entropy introduced in
arXiv:2206.09723 . We derive their holographic dual with the assumption that
the replica symmetry is unbroken in the bulk and check our prescription with
explicit computations in $2d$ CFTs. Analogous to the multi-entropy, the
holographic dual of these measures is given by the weighted area of the minimal
brane-web but with branes having differing tensions. We discuss the replica
symmetry assumption and also how the already known entanglement measures, such
as entanglement negativity and reflected entropy fit in our framework.
