
Symmetric Operads and Shuffle Operads 151
Let us describe an explicit construction of the free shuffle operad with a
given set of generators.
Definition 5.3.1.2 (Shuffle tree monomial). Let X = {X(n)}
n≥1
be
a reduced operation alphabet. A shuffle tree monomial in X is a triple
T = (τ, x, n), where
• τ is a planar rooted tree all of whose endpoints are leaves;
• x is a labelling of all internal vertices of τ by elements of X; each vertex
v must have a label x
v
∈ X(|Parent
−1
(v)|);
• n is a numbering of leaves of τ by integers 1, . . . , |Leaves(τ)| satisfying
the following local increasing condition stated as follows.
Any numbering n of leaves induces a numbering