  The downward closure of a language $L$ of words is the set of all (not
necessarily contiguous) subwords of members of $L$. It is well known that the
downward closure of any language is regular. Although the downward closure
seems to be a promising abstraction, there are only few language classes for
which an automaton for the downward closure is known to be computable.
  It is shown here that for stacked counter automata, the downward closure is
computable. Stacked counter automata are finite automata with a storage
mechanism obtained by \emph{adding blind counters} and \emph{building stacks}.
Hence, they generalize pushdown and blind counter automata.
  The class of languages accepted by these automata are precisely those in the
hierarchy obtained from the context-free languages by alternating two closure
operators: imposing semilinear constraints and taking the algebraic extension.
The main tool for computing downward closures is the new concept of Parikh
annotations. As a second application of Parikh annotations, it is shown that
the hierarchy above is strict at every level.
