Monomial identities in the Weyl algebra

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.