Samurai and the PARTITIONing problem

You are given an array $arr_1,arr_2...arr_n$ containing all the integers from $1$ to $n$ exactly once. We call a subarray $arr[i...j]$ of length greater than or equal to $2$, $left$ $heavy$, if the sum of all the elements of $arr[i...k-1]$ is strictly greater than that of $arr[k+1...j]$, where $arr_k$ is the smallest element of the subarray $arr[i...j]$. Note that if one of them is empty, we will