  Motivated by a question and some enumerative conjectures of Richard Stanley,
we explore the equivalence classes of words in the Weyl algebra, $\mathbf{k}
\langle D,U\rangle/(DU-UD=1)$. We show that each class is generated by the
swapping of adjacent *balanced subwords*, i.e., those which have the same
number of $D$'s as $U$'s, and give several other characterizations.
  Armed with this we deduce a number of enumerative results about the number of
such equivalence classes and their sizes. We extend these results to the class
of $c$-Dyck words, where every prefix has at least $c$ times as many $U$'s as
$D$'s. We also connect these results to previous work on bond percolation and
rook theory, and generalize them to some other algebras.
