iff $W_i= \{x\, | \,b(x)=a(x) +i \mod 3 \}.$ We have that $a(x_1 +\dots +x_k)=b(x_1 +\dots +x_k)$ (Here $ \,  a(x_1 +\dots +x_k)$ is the value of $ x_1 +\dots +x_k$ under $a$ (analogously for $b$).) iff $\sum_{i=0, 1, 2} \,i \cdot \sharp \{ j \,| \,x_j \in W_i\} \,= \,0 \mod 3.$ This is equivalent to $ \sharp \{ j \,| \,x_j \in W_1\}\,= \, \sharp \{ j \,| \,x_j \in W_2\} \mod 3.$ Given $\bar{l}= (l_0, l_1, l_2) $ with $\sum l_i \, =\, km $ we let ${ \cal K}(\bar{l})$ be the set of all $3\times m-$matrices $(k_{i, j})_{0\le i\le 2, 1\le j \le m}$ with $k_{1, j}= k_{2, j} \mod 3 $ and each column sums to $k$, that is $\sum_i k_{i, j}\,= k$ for each $j.$ Moreover, $ \sum_j k_{i,j} \,=\, l_i $ for $ i=0, 1, 2 $ ( the $i'$th row sums to $l_i.)$

We denote $$\begin{aligned}
 K(\bar{l}) :=
 \sum_{(k_{i, j})\in {\cal K}(\bar{l})}\,\prod_{j=1}^m{ k \choose k_{0, j}\,, k_{1, j}\,, k_{2, j}} .\, \, 
\mbox{ Then } \, \, \hat{N}(\bar{w}, \bar{l})\,:= \,  K(\bar{l}) \cdot \prod_{i=0}^2 M(l_i, w_i)  \label{NUFO}\end{aligned}$$ is the number of formulas $F$ true under two assignments $a, b$ with $b\,= \, D(a, 