Too Many LIS 2

*This problem has two subtasks worth $50$ points each.* You are given an integer $N$. You also have $N$ empty sets $S_1, S_2, \ldots, S_N$. Consider all the arrays $A$ of length $N$ such that $0 \leq A_i \leq 1$ for each $1 \leq i \leq N$ (there are $2^N$ such arrays in total). For each of these $2^N$ arrays, we compute an array $B$ of length $N$ where $B_i =$ length of longest non-decreasing s