
128 Algebraic Operads: An Algorithmic Companion
If T = (π, m) is a shuffle monomial of the same arity n
2
as T
2
, the
insertion operation replaces the part
p+k
1
−1
F
j=p
I
(j)
1
of π
1
by ˜π, the parti-
tion obtained from π by the bijection σ
−1
, and also replaces the subword
m
(p)
1
⊗m
(p+1)
1
⊗···⊗m
(p+k
1
−1)
1
of m
1
by m. Then, this operation is extended
by linearity to all shuffle polynomials of the same arity.
Remark 4.4.2.5. Our notation is not completely precise, since there may be
several different divisors T
2
inside T
1
. We always assume that the operation
T
1
,T
2
inserts everything at a particular occurrence of T
2
inside T
1
which is
implicit.
Example 4.4.2.6. Consider the sh