Constructing set-operads from monoids (Q447872)

Let \(M\) be a monoid. Let \(TM(n)\) be the collection of formal words of length \(n\) on the elements of \(M\). The author observes that the collection of these sets \(TM(n)\), \(n\in\mathbb N\), forms a symmetric operad naturally associated to \(M\). He then studies suboperads of the operad \(TM\) associated to the monoids \(M = \mathbb N^k\), \(k = 1,2,3\). He proves that usual word collections occurring in algebraic combinatorics (like endofunctions, parking functions, and other collections of words specified by growth constraints in \(\mathbb N\)), are identified with suboperads of these operads \(TM\).