Prefix Suffix LIS Counting

Let's denote the length of the Longest Increasing Subsequence of the array $B$ by $\mathbf{LIS}(B)$. For a permutation $A$ of integers from $1$ to $N$ we can define two arrays of integers $P$ and $S$, both of length $N$: $P_i = \mathbf{LIS}(A[1:i]) = \mathbf{LIS}([A_1, A_2, \ldots, A_i])$, $S_i = \mathbf{LIS}(A[i:N]) = \mathbf{LIS}([A_i, A_{i+1}, \ldots, A_N])$. Let's call a pair of arrays $(P,