A slightly different approach:
Code: Select all
\begin{align*}
N_{n,k}\triangleq
\Biggl\{ &\bigg[ X_0,X_1,\ldots,X_{k-2},S+\sum_{i=0}^{k-2}X_i\biggr]
\Biggm| \Biggr. \\
&\,[b_0, b_1,\ldots, b_{k-1}]\in N_k^{\prime 0}\cup N_k^{\prime 1},
\ X_i\in X_n^{b_i}, \\
\Biggl. &\, 0\leq i<k-1,\ S\in N_n^p,\ p=\sum_{i=0}^{k-1}b_i \Biggr\}.
\end{align*}
Please note that centered dots (\cdots) should mark ellipsis between binary relations or operators, not in enumerations. I've also added some spaces and removed some superfluous braces.