. We propose a [[[[**c**]{}]{}[[**d**]{}]{}]{}]{}-analogue to this conjecture; see Conjecture \[c:balance\] in Section \[s:balance\]. There is plenty of evidence as to why this conjecture should be true. Theorems \[t:adjoining\_balance\] and \[t:suff\_cond\] are important results in this direction.

5.  What can be said about formulas for the $\beta$ values ?

    We show that $\beta({{\bf c}}^i {{\bf d}}{{\bf c}}^j) = {\binom {i+j+2}{i+1}} -1$ and $\beta({{\bf d}}^n) = \frac{1}{2^n} E_{2n+1}$, where $E_{2n+1}$ are the Euler or tangent numbers (Example \[eg:compute\] and Lemma \[l:deuler\] respectively). Further data suggests that studying exact values in detail may be very interesting. For example, many of the $\beta$ values are divisible by 1001; see Section \[s:conc\_rmks\]. However, the thrust of this paper is on studying inequalities.

Organisation of the paper
-------------------------

We begin with the study of the polynomial algebra ${\mathcal F} = k \langle {{\bf c}},{{\bf d}}\rangle$ in Section \[s:algebra\]. Following Ehrenborg-Readdy , we define a coproduct $\Delta$ and a derivation $G$ on the algebra ${\mathcal F} = k \langle {{\bf c}},{{\bf d}}\rangle$. We first modify ${{\mathcal F}}$ to ${\hat{\mathcal F}}$ by adding a piece of degree $-1$ and then extend $\Delta$ and $G$ to ${\hat \Delta}$ and $\hat G$ respectively. The main result of Section \[s:algebra\] 