defined by }  Q(a)=\frac{m}{n} \nonumber\end{aligned}$$

Throughout we use $s=s(k, \gamma)$ uniquely defined by $Q(s) = k \gamma= k \gamma n/n=km/n.$ Note that for $k \ge 3$ we can assume that $k \gamma >2$ and $s$ always exists. We have $Q(s) \ge s.$ We often write $Q$ instead of $Q(s).$ Recall $N_0=M(km, n)$ and we get a tight bound on the number of formulas (cf. (\[DEFN0\]).)

\[N0\] $ N_0 \,= \, \Theta(1) \left(k \gamma n/(s e) \right)^{k \gamma n}\cdot q(s)^n. $

We treat the sum $ K(\bar{l})$ similarly to $M(m, n).$ Instead of $q(x)$ we use the function, $$\begin{aligned}
 r(\bar{x}):= \sum_{k_1 = k_2 \mod 3}  { k \choose k_0, k_1, k_2} x_0^{k_0}x_1^{k_1} x_2^{k_2}, \, \,  \bar{x}=(x_0, x_1, x_2).\, \mbox{ Then} \, \,\nonumber \\
{K(\bar{l})} \, =\, 
 \sum_{(k_{i, j})\in {\cal K}(\bar{l})}\,\prod_{j=1}^m{ k \choose k_{0, j}\,, k_{1, j}\,, k_{2, j}} \, \, = \, \mbox{Coeff}[\bar{x}^{\bar{l}},\, r(\bar{x})^m\,  ] \, 
< \,\frac{ r(\bar{c})^m}{\bar{c}^{\bar{l}}}   \label{K1}\end{aligned}$$ with the notation $ \bar{x}^{\bar{l}} = \prod_i x_i^{l_i}$ and $\bar{c}=(c_0, c_1, c_2)>0,$ meaning $c_i >0$ for all $i.$

For calculations it is useful to have a different representation of $r(\bar{x}).$ Let $\i$ be the imaginary unit, and ${\bf w_1}\,:= -1/2+ (\sqrt{3}/2) \i$ is the primitive third root of unity, ${ \bf 