  We give an explicit combinatorial formula for some irreducible components of
$GL_k\times \mathbb{S}_n$-modules of multivariate diagonal harmonics. To this
end we introduce a new path combinatorial object $T_{n,s}$ allowing us to give
the formula directly in terms of Schur functions. This paper also contains
formulas written in terms of Schur functions in the $q$ and $t$ variables for
special cases of $\nabla(e_n)$, $\nabla^r(e_n)$ and $\Delta'_{e_k}(e_n)$. We
also give an interpretation in term of path to the adjoint dual Pieri rule
applied on these $GL_k\times \mathbb{S}_n$-characters.
