Let's say we have to find the number of ways to partition a string into palindromic substrings of even length. We can solve this problem in $$$O(n log n)$$$ using Palindromic tree where we have to use an extended version of the palindromic tree consisting of series links. You can solve this beautiful problem using this idea.

But then I found out that if the question would have asked for the number of ways to partition a string into palindromic substrings of length $$$l$$$ such that $$$l \bmod p \lt = r$$$, then I couldn't solve it.

So here I am, asking you, is there any kind of generalized solution for this kind of problem? Maybe we can solve this problem for some small $$$p$$$?