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Motivated by a question and some enumerative conjectures of Richard Stanley, we explore the equivalence classes of words in the Weyl algebra, k D, U/ (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.
Grinberg et al. (Thu,) studied this question.
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