Letter most:

For each lowercase letter count number of its occurrences in $$$s$$$, maximum of this value for all letters is the answer. All can be done in $$$O(n)$$$.

code

Array modification:

Assume  will be the maximum. it can be proven that all operation should be done in direction of $$$x$$$ ($$$j=i+1$$$ for all operation on its left and $$$j=i-1$$$ for all operations on its right).

Now consider $pref_x$ is maximum value of $$$a_x$$$ if all operations be done in direction of right for all $i=1, 2, ..., x$. And consider $suf_x$ is maximum value of $$$a_x$$$ if all operations be done in direction of left for all $i=x, x+1, ..., n$.

It can bee seen that:

$$$pref_x = \frac{perf_{x-1}}{2} + a_x$$$.

$$$suf_x = \frac{suf_{x+1}}{2} + a_x$$$.

And so the answer is $max_{i=1}^{n}(pref_i + suf_i — a_i)$.

All can be done in $O(n)$.