  We study Jack polynomials in $N$ variables, with parameter $\alpha$, and
having a prescribed symmetry with respect to two disjoint subsets of variables.
For instance, these polynomials can exhibit a symmetry of type AS, which means
that they are anti-symmetric in the first $m$ variables and symmetric in the
remaining $N-m$ variables. One of our main goals is to extend recent works on
symmetric Jack polynomials [arXiv:0711.3062, arXiv:1007.2692, arXiv:1303.4126]
and prove that the Jack polynomials with prescribed symmetry also admit
clusters of size $k$ and order $r$, that is, the polynomials vanish to order
$r$ when $k+1$ variables coincide.
  We first prove some general properties for generic $\alpha$, such as their
uniqueness as triangular eigenfunctions of operators of Sutherland type, and
the existence of their analogues in infinity many variables. We then turn our
attention to the case with $\alpha=-(k+1)/(r-1)$. We show that for each triplet
$(k,r,N)$, there exist admissibility conditions on the indexing sets, called
superpartitions, that guaranty both the regularity and the uniqueness of the
polynomials. These conditions are also used to establish similar properties for
non-symmetric Jack polynomials. As a result, we prove that the Jack polynomials
with arbitrary prescribed symmetry, indexed by $(k,r,N)$-admissible
superpartitions, admit clusters of size $k=1$ and order $r\geq 2$. In the last
part of the article, we find necessary and sufficient conditions for the
invariance under translation of the Jack polynomials with prescribed symmetry
AS. This allows to find special families of superpartitions that imply the
existence of clusters of size $k>1$ and order $r\geq 2$.
