algebra $k \langle {{\bf c}},{{\bf d}}\rangle$ {#s:algebra}
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The basic algebraic object to consider is ${\mathcal A} = k \langle {{\bf a}},{{\bf b}}\rangle$, the free algebra in two non-commuting variables [**a**]{} and [**b**]{}. The other object of interest is the subalgebra ${\mathcal F}$, generated by the elements ${\bf c} := {{\bf a}}+ {{\bf b}}$ and ${\bf d} := {{{\bf a}}{{\bf b}}}+ {{\bf b}}{{\bf a}}$. Since we are primarily interested in the ${{{\bf c}}{{\bf d}}}$-index, we will concentrate on ${{\mathcal F}}$ and never deal with $\mathcal A$.

The basic setup
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We begin by recalling some facts from . Let $k$ be a field of characteristic 0. Let ${\mathcal F} = k \langle {{\bf c}},{{\bf d}}\rangle$ be the polynomial algebra in the non-commuting variables [**c**]{} and [**d**]{}. By setting the degree of [**c**]{} to be 1 and of [**d**]{} to be $2$, we write ${\mathcal F} = \oplus_{n \geq 0} {\mathcal F}_n$, where ${\mathcal F}_n$ is spanned by the [**cd**]{}-monomials of degree $n$. The product in ${{\mathcal F}}$ is given by concatenation and the unit element is 1.

\[p:d\] The vector space ${\mathcal F}$ has a (coassociative) coproduct $\Delta \colon \mathcal F \rightarrow \mathcal F {\otimes}\mathcal F$ given by the initial 