Enumerating Multi-Operator Monomials in Commutative and Noncommutative Settings

Abstract

We study enumeration problems for multi-operator monomials generated from one indeterminate by an associative multiplication together with finitely many unary operators. We consider four regimes, according to whether multiplication is commutative and whether the unary operators commute. In the case where the unary operators do not commute, we obtain explicit multigraded generating functions and coefficient formulas, including a multinomial refinement of the Narayana numbers, together with interpretations in terms of rooted trees, restricted lattice paths, and binary trees. When the unary operators commute, we derive canonical representatives and effective recurrences, with corresponding monotonicity conditions in the combinatorial models. When multiplication is commutative, the sequence decomposition is replaced by a multiset decomposition, leading to exp--log generating functions and Euler-transform recurrences. In special cases, the resulting sequences recover classical families including the Catalan numbers, the small Schr\"oder numbers, and rooted-tree numbers.