# Sigma Summation - Several Lower Indices

I was taking a look to a book of statistical mechanics, many equations show something as follows: $$Q(K,N) = \Sigma_{s_{1},s_{2},...,s_{N}=\pm 1}[ e^{K(...+s_1s_2+s_2s_3+s_3s_4...)} ]$$ then they partition the sum as follows $$Q(K,N) = \Sigma_{s_{1},s_{2},...,s_{N}}e^{K(s_1s_2+s_2s_3)}e^{K(s_3s_4+s_4s_5)} ...$$ After Summing over even numbered S's $$Q(K,N) = \Sigma_{s_{odd}} (e^{K(s_1+s_3)}+e^{-K(s_1+s_2)})(e^{K(s_3+s_5)}+e^{-K(s_3+s_5)})$$ How can I interpret this Summation with many lower indices.

$$Q(K,N) = \Sigma_{s_{1},s_{2},...,s_{N}=\pm 1}[ e^{K(...+s_1s_2+s_2s_3+s_3s_4...)} ]$$
To me, it looks like the summation is over the $$2^N$$ sets $$\{s_{1},s_{2},...,s_{N}\}$$ where each $$s_i$$ is either $$1$$ or $$-1$$.
For example, for $$N=2$$, these are
$$\{1, 1\}, \{1, -1\}, \{-1, 1\}, \{-1, -1\}$$.
• So would we sum in the following way: $$\Sigma_{S_1} F(S_i)=F(1) + F(-1)$$ and $$\Sigma_{S_1,S_2,...} F(S_i)=(F(1) + F(-1))+(F(1) + F(-1))+...$$