For the Boltzmann distribution, detailed balance results in $$\begin{equation*} \frac{w_i}{w_j}= \exp{(-\beta(E_i-E_j))}. \end{equation*}$$

Let us now specialize to a system whose energy is defined by the orientation of single spins. Consider the state $i$, with given energy $E_i$ represented by the following $N$ spins $$\begin{equation*} \begin{array}{cccccccccc} \uparrow&\uparrow&\uparrow&\dots&\uparrow&\downarrow&\uparrow&\dots&\uparrow&\downarrow\\ 1&2&3&\dots& k-1&k&k+1&\dots&N-1&N\end{array} \end{equation*}$$