  Indexed languages are a classical notion in formal language theory. As the
language equivalent of second-order pushdown automata, they have received
considerable attention in higher-order model checking. Unfortunately, counting
properties are notoriously difficult to decide for indexed languages: So far,
all results about non-regular counting properties show undecidability.
  In this paper, we initiate the study of slice closures of (Parikh images of)
indexed languages. A slice is a set of vectors of natural numbers such that
membership of $u,u+v,u+w$ implies membership of $u+v+w$. Our main result is
that given an indexed language $L$, one can compute a semilinear representation
of the smallest slice containing $L$'s Parikh image.
  We present two applications. First, one can compute the set of all affine
relations satisfied by the Parikh image of an indexed language. In particular,
this answers affirmatively a question by Kobayashi: Is it decidable whether in
a given indexed language, every word has the same number of $a$'s as $b$'s.
  As a second application, we show decidability of (systems of) word equations
with rational constraints and a class of counting constraints: These allow us
to look for solutions where a counting function (defined by an automaton) is
not zero. For example, one can decide whether a word equation with rational
constraints has a solution where the number of occurrences of $a$ differs
between variables $X$ and $Y$.
