Let's denote dp[r] = minimum cost to partition the first r elements of nums. What would be the transitions of such dynamic programming?

dp[r] = min(dp[l] + importance(nums[l..r])) over all 0 <= l < r. This already gives us an O(n^3) approach, as importance can be calculated in linear time, and there are a total of O(n^2) transitions.

Can you think of a way to compute multiple importance values of related subarrays faster?

importance(nums[l-1..r]) is either importance(nums[l..r]) if a new unique element is added, importance(nums[l..r]) + 1 if an old element that appeared at least twice is added, or importance(nums[l..r]) + 2, if a previously unique element is duplicated. This allows us to compute importance(nums[l..r]) for all 0 <= l < r in O(n) by keeping a frequency table and decreasing l from r-1 down to 0.