  We study counting-regular languages -- these are languages $L$ for which
there is a regular language $L'$ such that the number of strings of length $n$
in $L$ and $L'$ are the same for all $n$. We show that the languages accepted
by unambiguous nondeterministic Turing machines with a one-way read-only input
tape and a reversal-bounded worktape are counting-regular. Many one-way
acceptors are a special case of this model, such as reversal-bounded
deterministic pushdown automata, reversal-bounded deterministic queue automata,
and many others, and therefore all languages accepted by these models are
counting-regular. This result is the best possible in the sense that the claim
does not hold for either $2$-ambiguous PDA's, unambiguous PDA's with no
reversal-bound, and other models.
  We also study closure properties of counting-regular languages, and we study
decidability problems in regards to counting-regularity. For example, it is
shown that the counting-regularity of even some restricted subclasses of PDA's
is undecidable. Lastly, $k$-slender languages -- where there are at most $k$
words of any length -- are also studied. Amongst other results, it is shown
that it is decidable whether a language in any semilinear full trio is
$k$-slender.
