Extension: Finite Telescoping Sums and Products

For a permutation $\pi$ of the integers from $1$ to $10$, define $$S(\pi) = \sum_{i=1}^{9} (\pi(i) - \pi(i+1))\cdot (4 + \pi(i) + \pi(i+1)),$$ where $\pi (i)$ denotes the $i$th element of the permutation. Suppose that $M$ is the maximum possible value of $S(\pi)$ over all permutations $\pi$ of the integers from $1$ to $10$. Determine the number of permutations $\pi$ for which $S(\pi) = M$.

Source: 2012-2013 Winter Online Math Open