\; {\prod}_k^{\prime\prime} \gamma_{\ell,u}
\label{eqn6}            $$ where the primed product ${\prod}^\prime$ is over all the $\beta_{\ell,u}$ for transitions through positive elementary moves leaving the energy unchanged, and the double primed product ${\prod}^{\prime\prime}$ is over all the $\gamma_{\ell,u}$ where the transition is a positive elementary move increasing the energy. In this expression the $\ell$ and $u$ are functions of $n$, $m$, and $k$ in each of the products.

A weight $W(\phi_N)$ will be assigned to the sequence $\phi_N$. In order to compute the weight, define $$\fl
\sigma(j,j\plus 1) = \sigma(w_j\to w_{j+1}) = 
\cases{
-1, &\hbox{if $\Delta_j=+1$ and $\delta_j=0$}; \\
+1, &\hbox{if $\Delta_j=-1$ and $\delta_j=0$}; \\
\hspace{0.8em} 0, &\hbox{otherwise}. \\
}$$ Similarly, define $$\fl
\rho (j,j\plus 1) = \rho (w_j\to w_{j+1}) = 
\cases{
-1 , &\hbox{if $\Delta_j=+1$ and $\delta_j>0$}; \\
+1, &\hbox{if $\Delta_j=-1$ and $\delta_j<0$}; \\
\hspace{0.8em} 0, &\hbox{otherwise}. \\
}$$ That is, the function $\sigma(j,j\plus 1)$ tracks the negative and positive moves along $\phi_N$ where the energy is not changed, and the function $\rho(j,j\plus 1)$ tracks the negative and positive transitions along $\phi_N$ where the energy is also changed.

Assign the weight $$\fl
W(\phi_N) = \LB \Sfrac{\alpha^{--}_0 + \alpha^{-0}_0 + \alpha^{+0}_0 \beta_0 
+ \alpha^{++}_0 \gamma_0}{
\alpha^{--}_N +\alpha^{-0}_N +\alpha^{+0}_N \beta_N+ \alpha^{++}_N \gamma_N} \RB
