Séminaire Lotharingien de Combinatoire, 95B.75 (2026), 12 pp.
Darij Grinberg, Tom Roby, Stephan Wagner and Mei Yin
Downs and Ups in the Weyl Algebra
Abstract.
Motivated by a question and some 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. These allow us to 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 our results to previous work on rook theory.
Received: November 15, 2025.
Accepted: February 15, 2026.
Final version: April 1, 2026.
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