  We study (asymmetric) $U$-statistics based on a stationary sequence of
$m$-dependent variables; moreover, we consider constrained $U$-statistics,
where the defining multiple sum only includes terms satisfying some
restrictions on the gaps between indices. Results include a law of large
numbers and a central limit theorem. Special attention is paid to degenerate
cases where, after the standard normalization, the asymptotic variance
vanishes; in these cases non-normal limits occur after a different
normalization.
  The results are motivated by applications to pattern matching in random
strings and permutations. We obtain both new results and new proofs of old
results.
