The main idea for this problem is to compute the maximum number of pairs of identical elements which have the same "shift" value (e.g. $$$s[4]$$$ and $$$s[1]$$$ is an example of such a pair with shift $$$4 - 1 = 3$$$, since $$$s[4] = s[1]$$$). With this in mind, the trick is:

For each distinct character $$$c$$$, we will represent the set of that character's positions with a polynomial $$$f(x)$$$, where term $$$x^i$$$ exists if and only if $$$s[i] = c$$$. Then, we can imagine a second string with the same indices but negative represented by $$$g(x)$$$, since the indices of the second string are effectively "cancelling" that of the first. Once we multiply these 2 polynomials together, it will effectively produce a frequency array for each shift value. For example, in the pair mentioned above, we notice that $$$f(x)$$$ will contain term $$$x^4$$$ and $$$g(x)$$$ will contain term $$$x^{-1}$$$. From this specific pair, the resulting polynomial $$$f(x) \times g(x)$$$ will have the coefficient of term $$$x^4 \times x^{-1} = x^3$$$ increased by $$$1$$$.

Hopefully from the example above you see why the resulting polynomial $$$f(x) \times g(x)$$$ represents the frequency array of each shift value. Once you have the frequency array for each distinct character, you can simply add them up and find the terms with a positive degree having the largest coefficient.

The final complexity of this approach will be $$$O(3 \times N\log N)$$$ with FFT.