Given an integer array arr, you should partition the array into (contiguous) subarrays of length at most k. After partitioning, each subarray has their values changed to become the maximum value of that subarray.
Return the largest sum of the given array after partitioning.
 
Example 1:
Input: arr = [1,15,7,9,2,5,10], k = 3
Output: 84
Explanation: arr becomes [15,15,15,9,10,10,10]

Example 2:
Input: arr = [1,4,1,5,7,3,6,1,9,9,3], k = 4
Output: 83

Example 3:
Input: arr = [1], k = 1
Output: 1

 
Constraints:

1 <= arr.length <= 500
0 <= arr[i] <= 109
1 <= k <= arr.length

Difficulty: MEDIUM
Tags: Dynamic programming

Data structures
Skill types: Dynamic programming

Data structures

Starter code:
class Solution:
    def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
        

Solutions:
class Solution:

	def maxSumAfterPartitioning(self, arr, k):
		res = [0]
		for (idx, val) in enumerate(arr):
			(max_val, cur_val) = (0, 0)
			for i in range(max(0, idx - k + 1), idx + 1)[::-1]:
				if arr[i] > max_val:
					max_val = arr[i]
				if res[i] + (idx - i + 1) * max_val > cur_val:
					cur_val = res[i] + (idx - i + 1) * max_val
			res.append(cur_val)
		return res[-1]


class Solution:

	def maxSumAfterPartitioning(self, A: List[int], K: int) -> int:
		ans = [0]
		for (r, n) in enumerate(A):
			(maxVal, curVal) = (0, 0)
			for l in range(max(0, r - K + 1), r + 1)[::-1]:
				if A[l] > maxVal:
					maxVal = A[l]
				if ans[l] + (r - l + 1) * maxVal > curVal:
					curVal = ans[l] + (r - l + 1) * maxVal
			ans.append(curVal)
		return ans[-1]


class Solution:

	def maxSumAfterPartitioning(self, A: List[int], K: int) -> int:
		dp = [A[0]]
		for i in range(1, len(A)):
			max_partition_sum = A[i] + dp[-1]
			max_element = A[i]
			end = i - K
			if end < -1:
				end = -1
			for j in range(i - 1, end, -1):
				if A[j] > max_element:
					max_element = A[j]
				partition_sum = (i - j + 1) * max_element
				if j > 0:
					partition_sum += dp[j - 1]
				if partition_sum > max_partition_sum:
					max_partition_sum = partition_sum
			dp.append(max_partition_sum)
		return dp[-1]


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		return self.dp(arr, 0, k, {})

	def dp(self, arr, i, k, hashmap):
		if i >= len(arr):
			return 0
		if i in hashmap:
			return hashmap[i]
		largest = 0
		ans = 0
		for idx in range(k):
			if idx + i >= len(arr):
				break
			largest = max(largest, arr[i + idx])
			ans = max(ans, largest * (idx + 1) + self.dp(arr, i + idx + 1, k, hashmap))
		hashmap[i] = ans
		return ans


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		return self.recur(arr, len(arr) - 1, k, {})

	def recur(self, arr, index, k, hashMap):
		if index < 0:
			return 0
		if index in hashMap:
			return hashMap[index]
		result = 0
		maxV = 0
		for i in range(k):
			if index - i < 0:
				continue
			maxV = max(maxV, arr[index - i])
			result = max(result, maxV * (i + 1) + self.recur(arr, index - i - 1, k, hashMap))
		hashMap[index] = result
		return result


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		self.dp = {}

		def recur(index, arr, k):
			if index >= len(arr):
				return 0
			if index in self.dp:
				return self.dp[index]
			maxi = 0
			res = 0
			for i in range(k):
				if index + i < len(arr):
					maxi = max(maxi, arr[index + i])
					res = max(res, maxi * (i + 1) + recur(index + i + 1, arr, k))
			self.dp[index] = res
			return res
		return recur(0, arr, k)


class Solution:

	def util(self, i, n, A, K, d):
		if i >= n:
			return 0
		if i not in d:
			count = 0
			ma = A[i]
			temp = -1
			while i + count < n and count <= K - 1:
				if A[i + count] > ma:
					ma = A[i + count]
				nex = self.util(i + count + 1, n, A, K, d)
				if (count + 1) * ma + nex > temp:
					temp = (count + 1) * ma + nex
				count += 1
			d[i] = temp
		return d[i]

	def maxSumAfterPartitioning(self, A: List[int], K: int) -> int:
		n = len(A)
		d = {}
		return self.util(0, n, A, K, d)


class Solution:

	def maxSumAfterPartitioning(self, A: List[int], K: int) -> int:
		visited = {}

		def recurse(index):
			if index >= len(A):
				return 0
			if index in visited:
				return visited[index]
			mxx = 0
			ans = 0
			for i in range(K):
				if index + i < len(A):
					mxx = max(mxx, A[i + index])
					ans = max(ans, mxx * (i + 1) + recurse(i + index + 1))
			visited[index] = ans
			return ans
		return recurse(0)


class Solution:

	def maxSumAfterPartitioning(self, A: List[int], K: int) -> int:
		n = len(A)
		dp = [0] * (n + 1)
		for i in range(n):
			curMax = 0
			for k in range(1, min(K, i + 1) + 1):
				curMax = max(curMax, A[i - k + 1])
				dp[i] = max(dp[i], dp[i - k] + curMax * k)
		print(dp)
		return dp[n - 1]


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		dp = [0] * len(arr)
		dp[0] = arr[0]
		max_value = arr[0]
		max_sum = arr[0]
		for i in range(1, k):
			max_value = max(max_value, arr[i])
			dp[i] = max_value * (i + 1)
		for i in range(k, len(arr)):
			max_value = 0
			for j in range(k):
				max_value = max(max_value, arr[i - j])
				dp[i] = max(dp[i], dp[i - j - 1] + max_value * (j + 1))
		return dp[-1]


from collections import deque

class Solution:

	def maxSumAfterPartitioning(self, A: List[int], K: int) -> int:
		dp = deque([0 for i in range(K)])
		for i in range(1, len(A) + 1):
			(maxA, maxP) = (float('-inf'), float('-inf'))
			for j in range(1, min(K + 1, i + 1)):
				maxA = max(maxA, A[i - j])
				maxP = max(maxP, dp[-j] + j * maxA)
			dp.popleft()
			dp.append(maxP)
		return dp[-1]


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		inf = 1061109567
		dp = [[(-inf, -inf) for _ in range(k + 1)] for _ in range(2)]
		last_max = 0
		dp[0][0] = (0, 0)
		for i in range(1, len(arr) + 1):
			for j in range(1, k + 1):
				if j == 1:
					dp[i % 2][j] = (last_max + arr[i - 1], arr[i - 1])
					last_max = dp[i % 2][j][0]
				elif i >= j:
					(last_s, last_n) = dp[(i - 1) % 2][j - 1]
					n = max(last_n, arr[i - 1])
					s = last_s - last_n * (j - 1) + n * j
					dp[i % 2][j] = (s, n)
					last_max = max(last_max, s)
		return last_max


class Solution:

	def maxSumAfterPartitioning(self, A, K):
		N = len(A)
		dp = [0] * (N + 1)
		for i in range(N):
			curMax = 0
			for k in range(1, min(K, i + 1) + 1):
				curMax = max(curMax, A[i - k + 1])
				dp[i] = max(dp[i], dp[i - k] + curMax * k)
		return dp[N - 1]


class Solution:

	def maxSumAfterPartitioning(self, A: List[int], K: int) -> int:
		N = len(A)
		if K == 1 or N == 1:
			return sum(A)
		dp = [0] * (N + 1)
		dp[0] = 0
		for i in range(N):
			mv = A[i]
			dp[i + 1] = dp[i] + A[i]
			for j in range(1, min(i + 1, K)):
				mv = max(mv, A[i - j])
				dp[i + 1] = max(dp[i + 1], dp[i - j] + (j + 1) * mv)
		return dp[N]


class Solution:

	def maxSumAfterPartitioning(self, A: List[int], K: int) -> int:
		size = len(A)
		if K == 1:
			return sum(A)
		dp = [0] * (size + 1)
		for i in range(size):
			cur_max = A[i]
			block_size = 1
			while block_size <= K and i - block_size + 1 >= 0:
				cur_max = max(cur_max, A[i - block_size + 1])
				dp[i + 1] = max(dp[i + 1], dp[i - block_size + 1] + block_size * cur_max)
				block_size += 1
		return dp[size]


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		n = len(arr)
		dp = [[-1] * (n + 1) for _ in range(n + 1)]

		def maxsum(i=0):
			if i >= n:
				return 0
			elif dp[i][n] != -1:
				return dp[i][n]
			else:
				mx = 0
				for j in range(i, min(i + k, n)):
					mx = max(arr[j], mx)
					dp[i][j] = mx * (j - i + 1)
					dp[i][n] = max(dp[i][n], dp[i][j] + maxsum(j + 1))
				return dp[i][n]
		return maxsum()


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		dp = [0] * (n := len(arr))
		dp[0] = curr_max = arr[0]
		for i in range(1, k):
			curr_max = max(curr_max, arr[i])
			dp[i] = (i + 1) * curr_max
		for i in range(k, n):
			curr_max = arr[i]
			for j in range(k):
				curr_max = max(curr_max, arr[i - j])
				dp[i] = max(dp[i], dp[i - j - 1] + (j + 1) * curr_max)
		return dp[-1]


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], K: int) -> int:
		n = len(arr)
		dp = [0 for _ in range(n + 1)]
		for i in range(n):
			currMax = 0
			for k in range(1, min(K, i + 1) + 1):
				currMax = max(currMax, arr[i - k + 1])
				dp[i] = max(dp[i], dp[i - k] + currMax * k)
		return dp[n - 1]


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		if arr is None or len(arr) == 0:
			return 0
		dp = [0 for _ in range(len(arr))]
		for i in range(len(arr)):
			max_elem = arr[i]
			j = 1
			while j <= k and i - j + 1 >= 0:
				max_elem = max(max_elem, arr[i - j + 1])
				if i - j >= 0:
					dp[i] = max(dp[i], max_elem * j + dp[i - j])
				else:
					dp[i] = max(dp[i], max_elem * j)
				j += 1
		return dp[-1]


class Solution:

	def maxSumAfterPartitioning(self, A: List[int], K: int) -> int:
		n = len(A)
		dp = [0] * (n + 1)
		for (i, a) in enumerate(A):
			r = a
			for j in range(1, K + 1):
				r = max(r, A[i - j + 1] if i - j + 1 >= 0 else float('-inf'))
				if i - j + 1 >= 0:
					dp[i + 1] = max(dp[i + 1], dp[i - j + 1] + r * j)
		return dp[-1]


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		n = len(arr)
		self.dp = [-1 for i in range(n)]

		def helper(pos: int) -> int:
			if pos >= n:
				return 0
			if self.dp[pos] != -1:
				return self.dp[pos]
			current_max = arr[pos]
			ret = arr[pos] + helper(pos + 1)
			for i in range(1, k):
				pos2 = pos + i
				if pos2 >= n:
					break
				current_max = max(current_max, arr[pos2])
				ret = max(ret, current_max * (i + 1) + helper(pos2 + 1))
			self.dp[pos] = ret
			return ret
		return helper(0)


class Solution:

	def maxSumAfterPartitioning(self, A: List[int], K: int) -> int:
		n = len(A)
		maxSum = [0] * n
		for i in range(-1, -K - 1, -1):
			maxSum[i] = max(A[i:]) * -i
		for i in range(n - K - 1, -1, -1):
			ms = 0
			for k in range(1, K + 1):
				ms = max(ms, max(A[i:i + k]) * k + maxSum[i + k])
			maxSum[i] = ms
		return maxSum[0]


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:

		@lru_cache(None)
		def helper(index, start, mx):
			if index == len(arr):
				return (index - start) * mx
			if index - start + 1 <= k:
				return max(helper(index + 1, start, max(arr[index], mx)), (index - start) * mx + helper(index + 1, index, arr[index]))
			return (index - start) * mx + helper(index + 1, index, arr[index])
		return helper(0, 0, arr[0])


class Solution:

	def maxSumAfterPartitioning(self, A: List[int], K: int) -> int:
		dp = [0] * len(A)
		for i in range(K):
			dp[i] = max(A[:i + 1]) * (i + 1)
		for i in range(K, len(A)):
			dp[i] = max([dp[i - j] + max(A[i - j + 1:i + 1]) * j for j in range(1, K + 1)])
		return dp[-1]


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		dp = [[(0, 0)] * k for _ in range(len(arr) + 1)]
		for i in range(1, len(arr) + 1):
			for j in range(min(i, k)):
				if j == 0:
					dp[i][j] = max(((dp[i - 1][m][0] + arr[i - 1], arr[i - 1]) for m in range(min(i, k))))
				else:
					new_max = max(dp[i - 1][j - 1][1], arr[i - 1])
					dp[i][j] = (dp[i - 1][j - 1][0] - j * dp[i - 1][j - 1][1] + new_max * (j + 1), new_max)
		return max(dp[-1])[0]


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		n = len(arr)
		dp = [0] * n
		for i in range(n - 1, -1, -1):
			curr_max = arr[i]
			for j in range(0, k):
				index = min(n - 1, i + j)
				curr_max = max(curr_max, arr[index])
				if i + j + 1 >= n:
					dp[i] = (n - i) * curr_max
				else:
					dp[i] = max(dp[i], dp[i + j + 1] + (j + 1) * curr_max)
		return dp[0]


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		if not arr:
			return 0
		local_max = {(idx, idx + 1): arr[idx] for idx in range(len(arr))}
		for l in range(2, k + 1):
			for start in range(len(arr) - l + 1):
				local_max[start, start + l] = max(local_max[start, start + l - 1], local_max[start + 1, start + l])
		f = [0]
		for end in range(1, len(arr) + 1):
			_local = 0
			for start in range(max(0, end - k), end):
				_local = max(f[start] + local_max[start, end] * (end - start), _local)
			f.append(_local)
		return f[-1]


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		n = len(arr)
		memo = [0] * n
		for i in range(n):
			if i < k:
				memo[i] = (i + 1) * max(arr[:i + 1])
			else:
				memo[i] = max((memo[i - j - 1] + (j + 1) * max(arr[i - j:i + 1]) for j in range(k)))
		return memo[-1]


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		length = len(arr)
		dp = [0] * (length + 1)
		maxval = [[0] * length for _ in range(length)]
		for i in range(length):
			for j in range(i, length):
				if i == j:
					maxval[i][j] = arr[i]
				else:
					maxval[i][j] = max(maxval[i][j - 1], arr[j])
		for i in range(1, length + 1):
			temp = 0
			for t in range(1, k + 1):
				temp = max(temp, dp[i - t] + maxval[i - t][i - 1] * t)
			dp[i] = temp
		return dp[-1]


class Solution:

	def maxSumAfterPartitioning(self, A: List[int], K: int) -> int:
		dp = [0] * len(A)
		for i in range(K):
			dp[i] = (i + 1) * max(A[:i + 1])
		for i in range(K, len(A)):
			maximum = 0
			for j in range(1, K + 1):
				if i - j < 0:
					break
				temp = dp[i - j] + max(A[i - j + 1:i + 1]) * j
				if temp > maximum:
					maximum = temp
			dp[i] = maximum
		print(dp)
		return dp[-1]


class Solution:

	def maxSumAfterPartitioning(self, A: List[int], K: int) -> int:

		def dfs(i, j, max_num=0):
			if i == -1:
				return max_num * j
			if opt[i][j] == -1:
				(a, b) = (0, 0)
				if j < K:
					a = dfs(i - 1, j + 1, max(max_num, A[i]))
				b = j * max_num + dfs(i - 1, 1, A[i])
				opt[i][j] = max(a, b)
			return opt[i][j]
		n = len(A)
		opt = [[-1] * (K + 1) for _ in range(n)]
		return dfs(n - 1, 0)


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		l = len(arr)
		results = [0] * l
		for (idx, num) in enumerate(arr):
			if idx < k:
				results[idx] = max(arr[:idx + 1]) * (idx + 1)
			else:
				res = 0
				for i in range(1, k + 1):
					start = idx - i
					tmp = results[start] + max(arr[start + 1:idx + 1]) * i
					res = max(res, tmp)
				results[idx] = res
		return results[-1]


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		max_sums = [0] * len(arr)
		for i in range(len(arr)):
			if i <= k - 1:
				max_sums[i] = (i + 1) * max(arr[:i + 1])
			else:
				res = 0
				for j in range(1, k + 1):
					first = max_sums[i - j]
					second = j * max(arr[i - j + 1:i + 1])
					res = max(res, first + second)
				max_sums[i] = res
		return max_sums[-1]


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		maxx = arr[0]
		dp = []
		for (i, _) in enumerate(arr[:k]):
			maxx = max(arr[i], maxx)
			dp.append(maxx * (i + 1))
		for (i, _) in enumerate(arr[k:]):
			i += k
			max_sum = 0
			for j in range(k):
				max_sum = max(max_sum, dp[i - j - 1] + max(arr[i - j:i + 1]) * (j + 1))
			dp.append(max_sum)
		return dp[-1]


class Solution:

	def printm(self, arr):
		[print(arr[i]) for i in range(len(arr))]

	def maxSumAfterPartitioning(self, A, K):
		size = len(A)
		dp = [0] * size
		for i in range(K):
			dp[i] = max(A[:i + 1]) * (i + 1)
		for i in range(K, size):
			for j in range(1, K + 1):
				dp[i] = max(dp[i], dp[i - j] + max(A[i - j + 1:i + 1]) * j)
		return dp[-1]


class Solution:

	def maxSumAfterPartitioning(self, A: List[int], K: int) -> int:
		N = len(A)
		dp = [float('-inf')] * (N + 1)
		for i in range(1, N + 1):
			if i <= K:
				dp[i] = max(A[:i]) * i
			else:
				for j in range(1, K + 1):
					dp[i] = max(dp[i], dp[i - j] + max(A[i - j:i]) * j)
		return dp[-1]


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		M = [0 for i in range(len(arr))]
		M[0] = arr[0]
		for i in range(1, len(arr)):
			if i < k:
				M[i] = max(arr[:i + 1]) * (i + 1)
			else:
				for j in range(1, min(k + 1, i + 1)):
					M[i] = max(M[i], M[i - j] + max(arr[i - j + 1:i + 1]) * j)
		return M[-1]


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		rec = {}

		def helper(idx):
			if idx in rec:
				return rec[idx]
			if len(arr) - idx <= k:
				temp = max(arr[idx:]) * (len(arr) - idx)
				rec[idx] = temp
				return temp
			pre_max = float('-inf')
			temp = float('-inf')
			for j in range(idx, min(idx + k, len(arr))):
				pre_max = max(pre_max, arr[j])
				temp = max(temp, pre_max * (j - idx + 1) + helper(j + 1))
			rec[idx] = temp
			return temp
		return helper(0)


class Solution:

	def maxSumAfterPartitioning(self, A: List[int], K: int) -> int:

		def maxSum(i, mem):
			if i in mem:
				return mem[i]
			if len(A) - i <= K:
				mem[i] = max(A[i:]) * (len(A) - i)
				return mem[i]
			ans = -float('inf')
			for j in range(i + 1, i + 1 + K):
				if j > len(A):
					break
				ans = max(ans, max(A[i:j]) * (j - i) + maxSum(j, mem))
			mem[i] = ans
			return mem[i]
		mem = {}
		return maxSum(0, mem)


class Solution:

	def maxSumAfterPartitioning(self, A: List[int], K: int) -> int:
		largest_sums = []
		for i in range(len(A)):
			if i < K:
				largest_sums.append(max(A[:i + 1]) * (i + 1))
				continue
			max_sum = 0
			for j in range(K):
				if i - (j + 1) < 0:
					continue
				cur_sum = largest_sums[i - j - 1] + max(A[i - j:i + 1]) * (j + 1)
				if cur_sum > max_sum:
					max_sum = cur_sum
			largest_sums.append(max_sum)
		return largest_sums[-1]


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		mm = [[a for a in arr]]
		for i in range(len(arr) - 1):
			mm.append([0] * len(arr))
		for i in range(1, k):
			for j in range(i, len(arr)):
				mm[i][j] = max(mm[i - 1][j - 1], arr[j])
		dp = [0] * (len(arr) + 1)
		for i in range(1, len(arr) + 1):
			for j in range(max(k, i)):
				dp[i] = max(dp[i], dp[i - j - 1] + mm[j][i - 1] * (j + 1))
		print(dp)
		return dp[-1]


class Solution:

	def maxSumAfterPartitioning(self, A: List[int], K: int) -> int:
		aLen = len(A)
		dp = {}

		def helper(i0):
			if i0 in dp:
				return dp[i0]
			if aLen - i0 <= K:
				dp[i0] = max(A[i0:]) * (aLen - i0)
				return dp[i0]
			subAns = 0
			thisMax = A[i0]
			for ki in range(1, K + 1):
				thisMax = max(thisMax, A[i0 + ki - 1])
				subAns = max(subAns, thisMax * ki + helper(i0 + ki))
			dp[i0] = subAns
			return subAns
		return helper(0)


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		n = len(arr)
		maxes = [[None for _ in range(n)] for _ in range(n)]
		for i in range(n):
			maxes[i][i] = arr[i]
		for i in range(n):
			for j in range(i + 1, n):
				maxes[i][j] = max(arr[j], maxes[i][j - 1])
		dp = [None for _ in range(n + 1)]
		dp[n] = 0
		for i in range(n - 1, -1, -1):
			m = float('-inf')
			for j in range(min(n - i, k)):
				m = max(m, (j + 1) * maxes[i][min(n - 1, i + j)] + dp[min(n, i + j + 1)])
			dp[i] = m
		return dp[0]


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:

		def helper(index, start, mx):
			key = (index, start, mx)
			if key in d:
				return d[key]
			if index == len(arr):
				return (index - start) * mx
			if index - start + 1 <= k:
				d[key] = max(helper(index + 1, start, max(arr[index], mx)), (index - start) * mx + helper(index + 1, index, arr[index]))
				return d[key]
			d[key] = (index - start) * mx + helper(index + 1, index, arr[index])
			return d[key]
		d = dict()
		return helper(0, 0, arr[0])


class Solution:

	def maxSumAfterPartitioning(self, A: List[int], K: int) -> int:

		@lru_cache(None)
		def checkSum(i, j, k):
			if j - i <= k:
				return max(A[i:j]) * (j - i)
			else:
				best = 0
				for nk in range(1, K + 1):
					best = max(best, checkSum(i, i + nk, K) + checkSum(i + nk, j, K))
			return best
		return checkSum(0, len(A), K)


class Solution:

	def maxSumAfterPartitioning(self, A: List[int], K: int) -> int:
		N = len(A)
		arr = [[(0, 0)] * (K + 1) for _ in range(N)]
		prev = 0
		for i in range(N):
			arr[i][1] = (A[i] + prev, A[i])
			prev += A[i]
			for j in range(2, min(i + 2, K + 1)):
				mx = max(arr[i - 1][j - 1][1], A[i])
				s = arr[i - 1][j - 1][0] - arr[i - 1][j - 1][1] * (j - 1) + mx * j
				arr[i][j] = (s, mx)
				prev = max(prev, s)
		return max((arr[-1][k][0] for k in range(1, K + 1)))


class Solution:

	def maxSumAfterPartitioning(self, A: List[int], K: int) -> int:
		if not A:
			return 0
		N = len(A)
		if K == N:
			return N * max(A)
		dp = [0 for i in range(N)]
		for i in range(K):
			dp[i] = max(A[:i + 1]) * (i + 1)
		for i in range(K, N):
			a = max(A[i - K + 1:i + 1])
			dp[i] = max([dp[i - j] + j * max(A[i - j + 1:i + 1]) for j in range(1, K + 1)])
		return dp[-1]


class Solution:

	def maxSumAfterPartitioning(self, A: List[int], K: int) -> int:

		@lru_cache(None)
		def help(A, K):
			n = len(A)
			if K == 1:
				return sum(A)
			if K >= n:
				return n * max(A)
			cMax = [A[0]]
			for i in range(1, K):
				cMax.append(max(cMax[-1], A[i]))
			return max(((i + 1) * cMax[i] + help(A[i + 1:], K) for i in range(K)))
		return help(tuple(A), K)


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		grid = [0] * len(arr)
		for i in range(len(arr) - 1, -1, -1):
			if len(arr) - i <= k:
				grid[i] = max(arr[i:len(arr)]) * (len(arr) - i)
			else:
				maxi = 0
				current_max = 0
				for t in range(k):
					current_max = max(current_max, arr[i + t])
					maxi = max(maxi, current_max * (t + 1) + grid[i + t + 1])
				grid[i] = maxi
		return grid[0]


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		if not arr:
			return 0
		if k == 1:
			return sum(arr)
		self.memo = {}

		def makepart(arr, n):
			if n in self.memo:
				return self.memo[n]
			if not arr:
				return 0
			l = min(k, len(arr))
			maxele = 0
			total = 0
			for i in range(l):
				maxele = max(maxele, arr[i])
				temp = maxele * (i + 1) + makepart(arr[i + 1:], n + i + 1)
				total = max(total, temp)
			self.memo[n] = total
			return total
		return makepart(arr, 0)


class Solution:

	def maxSumAfterPartitioning(self, A: List[int], k: int) -> int:

		def dfs(i, j, max_num=0):
			if i == -1:
				return max_num * j
			if (i, j, max_num) not in opt:
				ans = 0
				if j < k:
					ans = max(ans, dfs(i - 1, j + 1, max(max_num, A[i])))
				ans = max(ans, j * max_num + dfs(i - 1, 1, A[i]))
				opt[i, j, max_num] = ans
			return opt[i, j, max_num]
		n = len(A)
		opt = dict()
		return dfs(n - 1, 0)


class Solution:

	def maxSumAfterPartitioning(self, arr, k: int) -> int:
		n = len(arr)
		mask = [[0] * n for i in range(n)]
		for i in range(n):
			for j in range(i, n):
				mask[i][j] = max(arr[j], mask[i][j - 1] if j > i else 0)
		for i in range(n):
			for j in range(i, n):
				mask[i][j] = (j + 1 - i) * mask[i][j]
		dp = [0] * n
		for i in range(n):
			for h in range(1, k + 1):
				dp[i] = max((dp[i - h] if i - h >= 0 else 0) + (mask[i - h + 1][i] if i - h + 1 < n else 0), dp[i])
		return dp[-1]


class Solution:

	def maxSumAfterPartitioning(self, A: List[int], K: int) -> int:
		self.answer = {}

		def max_starting_i(index_A):
			array = A[index_A:]
			if index_A in self.answer:
				return self.answer[index_A]
			if len(array) == 0:
				self.answer[index_A] = 0
				return 0
			if len(array) <= K:
				self.answer[index_A] = max(array) * len(array)
				return self.answer[index_A]
			max_start_here = 0
			for i in range(0, K):
				max_split_here = (i + 1) * max(array[:i + 1]) + max_starting_i(index_A + i + 1)
				max_start_here = max(max_start_here, max_split_here)
			self.answer[index_A] = max_start_here
			return max_start_here
		return max_starting_i(0)


class Solution:

	def maxSumAfterPartitioning(self, l: List[int], k: int) -> int:

		def _max_sum(l, start, k, memo):
			if len(l) - start <= k:
				return (len(l) - start) * max(l[start:])
			if start in memo:
				return memo[start]
			max_sum = l[start]
			for i in range(1, k + 1):
				curr_sum = max(l[start:start + i]) * i + _max_sum(l, start + i, k, memo)
				max_sum = max(max_sum, curr_sum)
			memo[start] = max_sum
			return max_sum
		if not l:
			return 0
		return _max_sum(l, 0, k, dict())


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		self.dp = [-1] * (len(arr) + k)
		return self.solve(arr, 0, k)

	def solve(self, arr, start, k):
		if start >= len(arr):
			return 0
		if start + k >= len(arr):
			end = min(len(arr), start + k)
			return max(arr[start:end]) * (end - start)
		if self.dp[start] != -1:
			return self.dp[start]
		result = float('-inf')
		for p in range(1, k + 1):
			temp = max(arr[start:start + p]) * p + self.solve(arr, start + p, k)
			result = max(temp, result)
		self.dp[start] = result
		return result


class Solution:

	def partition(self, arr, k, index, memo):
		if len(arr) == 0:
			return 0
		if index in memo:
			return memo[index]
		maxNum = arr[0]
		maxSum = maxNum
		for i in range(1, min(k + 1, len(arr) + 1)):
			maxNum = max(maxNum, arr[i - 1])
			subarr = arr[:i]
			sumArr = self.partition(arr[i:], k, index + i, memo)
			sumArr += maxNum * i
			maxSum = max(sumArr, maxSum)
		memo[index] = maxSum
		return maxSum

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		memo = {}
		num = self.partition(arr, k, 0, memo)
		print(memo)
		return num


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		self.cache = {}

		def helper(arr, k, start):
			if start in self.cache:
				return self.cache[start]
			_arr = arr[start:]
			if len(_arr) == 0:
				return 0
			if len(_arr) <= k:
				return max(_arr) * len(_arr)
			ans = -1
			for i in range(1, k + 1):
				ans = max(ans, i * max(_arr[:i]) + helper(arr, k, start + i))
			self.cache[start] = ans
			return ans
		return helper(arr, k, 0)


import typing

class Solution:

	def maxSumAfterPartitioning(self, arr: typing.List[int], k: int) -> int:
		storage = {}

		def getSumRec(arr: list, i: int, k: int) -> int:
			if i in storage:
				return storage[i]
			if not i < len(arr):
				return 0
			if len(arr) - i < k:
				return max(arr[i:]) * len(arr[i:])
			listOfSums = []
			for offset in range(k):
				listOfSums.append(max(arr[i:i + offset + 1]) * (offset + 1) + getSumRec(arr, i + offset + 1, k))
			storage[i] = max(listOfSums)
			return storage[i]
		return getSumRec(arr, 0, k)


class Solution:

	def maxSumAfterPartitioning(self, A: List[int], K: int) -> int:
		best_a = [0] * len(A)
		best_a[0] = A[0]
		for i in range(1, len(A)):
			max_v = 0
			for j in range(i, max(-1, i - K), -1):
				sa = A[j:i + 1]
				v = best_a[j - 1] + max(sa) * len(sa)
				if v > max_v:
					max_v = v
			best_a[i] = max_v
		return best_a[-1]


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		dp = [0] * (len(arr) + 1)
		for i in range(0, len(arr)):
			m = 0
			for j in range(i, i - k, -1):
				if j < 0:
					break
				t = max(arr[j:i + 1]) * (i - j + 1) + dp[j]
				if t > m:
					m = t
			dp[i + 1] = m
		return dp[-1]


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		n = len(arr)
		dp_sum = [0] * n
		dp_sum[0] = arr[0]
		max_so_far = arr[0]
		for i in range(1, k):
			max_so_far = max(max_so_far, arr[i])
			dp_sum[i] = (i + 1) * max_so_far
		for i in range(k, n):
			partition_max = 0
			for back in range(k):
				partition_max = max(partition_max, arr[i - back])
				dp_sum[i] = max(dp_sum[i], dp_sum[i - back - 1] + (back + 1) * partition_max)
		return dp_sum[-1]


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		opt = [arr[0]]
		for i in range(1, len(arr)):
			opt.append(0)
			subseqLenMax = min(k, i + 1)
			for subseqLen in range(1, subseqLenMax + 1):
				subseqSum = subseqLen * max(arr[i - subseqLen + 1:i + 1])
				if subseqLen < i + 1:
					prevOpt = opt[i - subseqLen]
				else:
					prevOpt = 0
				optTemp = prevOpt + subseqSum
				if optTemp > opt[i]:
					opt[i] = optTemp
		return opt[len(opt) - 1]


class Solution:

	def maxSumAfterPartitioning(self, A: List[int], K: int) -> int:
		n = len(A)
		dp = [0] * (n + 1)
		for i in range(1, n + 1):
			for w in range(1, K + 1):
				if i - w < 0:
					break
				dp[i] = max(dp[i], dp[i - w] + max(A[i - w:i]) * w)
		return dp[-1]


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		dp = [0] * (len(arr) + 1)
		dp[1] = arr[0]
		for i in range(2, len(dp)):
			for j in range(1, k + 1):
				if i - j < 0:
					continue
				dp[i] = max(dp[i], dp[i - j] + j * max(arr[i - j:i]))
		print(dp)
		return dp[len(arr)]


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		n = len(arr)
		if not arr:
			return 0
		if n == 1:
			return arr[0]
		if n < k:
			return sum(arr)
		dp = [0] * n
		for i in range(n):
			for j in range(i, max(-1, i - k), -1):
				dp[i] = max(dp[i], max(arr[j:i + 1]) * (i - j + 1) + (dp[j - 1] if j - 1 >= 0 else 0))
		return dp[-1]


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		N = len(arr)
		T = [0] * (N + 1)
		for i in range(1, N + 1):
			for j in range(1, k + 1):
				if j > i:
					break
				T[i] = max(T[i], T[i - j] + max(arr[i - j:i]) * j)
		return T[N]


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		if k == 1:
			return sum(arr)
		if k == len(arr):
			max_val = max(arr)
			return max_val * len(arr)
		sums = [-1 for index in range(k)]
		maxs = [-1 for index in range(k)]
		max_sum = 0
		sums[0] = arr[0]
		maxs[0] = arr[0]
		for idx in range(1, len(arr)):
			val = arr[idx]
			max_sum = max(sums)
			for ki in range(k - 1, 0, -1):
				max_val = maxs[ki - 1]
				if not max_val < 0:
					if val <= max_val:
						maxs[ki] = max_val
						sums[ki] = sums[ki - 1] + max_val
					else:
						maxs[ki] = val
						sums[ki] = sums[ki - 1] - max_val * ki + val * (ki + 1)
			sums[0] = max_sum + val
			maxs[0] = val
		return max(sums)


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		dp = [0] * len(arr)
		for i in range(len(arr)):
			maxsum = 0
			for j in range(k):
				if i - j >= 1:
					maxsum = max(maxsum, (j + 1) * max(arr[i - j:i + 1]) + dp[i - j - 1])
				elif i - j >= 0:
					maxsum = max(maxsum, (j + 1) * max(arr[i - j:i + 1]))
			dp[i] = maxsum
		return dp[-1]


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		n = len(arr)
		dp = [float('-inf') for _ in range(n + 1)]
		dp[0] = 0
		for i in range(1, n + 1):
			j = i - 1
			cnt = k
			while j >= 0 and cnt >= 1:
				maximum = max(arr[j:i])
				dp[i] = max(dp[i], dp[j] + maximum * (i - j))
				j -= 1
				cnt -= 1
		return dp[-1]


class Solution:

	def maxSumAfterPartitioning(self, A: List[int], K: int) -> int:
		dp = [0] * (len(A) + 1)
		for i in range(1, len(A) + 1):
			temp = []
			for j in range(1, K + 1):
				if i - j >= 0:
					k = dp[i - j] + max(A[i - j:i]) * j
					temp.append(k)
			dp[i] = max(temp)
		return dp[len(A)]


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		dp = [0 for _ in range(k + 1)]
		rolling_max = -1
		for i in range(0, k):
			rolling_max = max(rolling_max, arr[i])
			dp[i] = rolling_max * (i + 1)
		for i in range(k, len(arr)):
			rolling_max = arr[i]
			for j in range(1, k + 1):
				rolling_max = max(rolling_max, arr[i - j + 1])
				dp[i % (k + 1)] = max(rolling_max * j + dp[(i - j) % (k + 1)], dp[i % (k + 1)])
		return dp[(len(arr) - 1) % (k + 1)]


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		dp = [0 for num in arr]
		for i in range(len(arr)):
			prev_sum = 0 if i - 1 < 0 else dp[i - 1]
			max_at_i = arr[i] + prev_sum
			possible_ks = i if i < k - 1 else k - 1
			for j in range(1, possible_ks + 1):
				current_window = arr[i - j:i + 1]
				current_max = max(current_window)
				current_window_sum = len(current_window) * current_max
				prev_window_sum = 0 if i - j - 1 < 0 else dp[i - j - 1]
				total_sum = current_window_sum + prev_window_sum
				if total_sum > max_at_i:
					max_at_i = total_sum
			dp[i] = max_at_i
		return dp[-1]


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		L = len(arr)
		record = dict()
		record[0] = 0
		for idx in range(1, L + 1):
			r = -float('inf')
			for gap in range(1, k + 1):
				if idx - gap < 0:
					continue
				else:
					r = max(r, max(arr[idx - gap:idx]) * gap + record[idx - gap])
			record[idx] = r
		return record[L]


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		hashmap = {}

		def partitionMatrix(lst, currScore):
			key = tuple(lst)
			if key in hashmap:
				return hashmap[key] + currScore
			if lst == []:
				hashmap[key] = 0
				return currScore
			for i in range(1, k + 1):
				if len(lst) == i:
					hashmap[key] = max(lst) * i
					return currScore + max(lst) * i
			best = currScore
			for i in range(1, k + 1):
				subScore = max(lst[:i]) * i
				best = max(best, partitionMatrix(lst[i:], currScore + subScore))
			hashmap[key] = best - currScore
			return best
		return partitionMatrix(arr, 0)


class Solution:

	def maxSumAfterPartitioning(self, A: List[int], K: int) -> int:
		dp = [[0 for _ in range(len(A))] for _ in range(2)]
		dp2 = [0 for _ in range(len(A))]
		for i in range(len(A)):
			startIndex = max(0, i - K + 1)
			while startIndex <= i:
				if startIndex - 1 >= 0:
					dp2[i] = max(dp2[startIndex - 1] + max(A[startIndex:i + 1]) * (i + 1 - startIndex), dp2[i])
				else:
					dp2[i] = max(A[:i + 1]) * (i + 1)
				startIndex += 1
		print(dp2)
		return dp2[-1]


class Solution:

	def maxSumAfterPartitioning(self, A: List[int], K: int) -> int:

		def rec(l, h):
			if h - l < K:
				dp[l][h] = (h - l + 1) * max(A[l:h + 1])
				return dp[l][h]
			m = []
			if dp[l][h] > 0:
				return dp[l][h]
			for i in range(0, K):
				m += [rec(l, l + i) + rec(l + i + 1, h)]
			dp[l][h] = max(m)
			return dp[l][h]
		dp = [0] * len(A)
		dp = [[0] * len(A) for i in range(len(A))]
		rec(0, len(A) - 1)
		return dp[0][len(A) - 1]


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		size = len(arr)
		mem = [0] * size
		for i in range(size):
			mem[i] = arr[i]
			for j in range(0, k):
				if i - j > 0:
					candidate = max(arr[i - j:i + 1]) * (j + 1) + mem[i - j - 1]
					mem[i] = max(mem[i], candidate)
				elif i - j == 0:
					candidate = max(arr[i - j:i + 1]) * (j + 1)
					mem[i] = max(mem[i], candidate)
				else:
					break
		return mem[size - 1]


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		dp = [0]
		for i in range(len(arr)):
			max_sum = -float('inf')
			for j in range(1, k + 1):
				if i - j + 1 < 0:
					break
				max_sum = max(max_sum, max(arr[i - j + 1:i + 1]) * j + dp[i - j + 1])
			dp.append(max_sum)
		return dp[-1]


class Solution:

	def getMax(self, arr, k, idx, cache):
		if idx == len(arr) - 1:
			return arr[idx]
		maxSum = 0
		for numInPartition in range(1, k + 1):
			if idx + numInPartition > len(arr):
				break
			startOfRecursiveIndex = idx + numInPartition
			maxVal = max(arr[idx:startOfRecursiveIndex])
			partSum = maxVal * numInPartition
			rest = cache[startOfRecursiveIndex] if startOfRecursiveIndex in cache else self.getMax(arr, k, startOfRecursiveIndex, cache)
			cache[startOfRecursiveIndex] = rest
			maxSum = max(maxSum, partSum + rest)
		return maxSum

	def maxSumAfterPartitioning(self, A: List[int], K: int) -> int:
		cache = {}
		return self.getMax(A, K, 0, cache)


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		mem = {}

		def find(s, e):
			if (s, e) in mem:
				return mem[s, e]
			if e - s == 1:
				return arr[s]
			elif e - s == 0:
				return 0
			else:
				m = None
				for i in range(s + 1, min(s + k + 1, len(arr) + 1)):
					subsum = max(arr[s:i]) * (i - s) + find(i, e)
					if m is None or subsum > m:
						m = subsum
				mem[s, e] = m
				return m
		return find(0, len(arr))


class Solution:

	def maxSumAfterPartitioning(self, A: List[int], K: int) -> int:
		from functools import lru_cache

		@lru_cache(None)
		def dp(i):
			ans = 0
			for k in range(K):
				if i - k >= 0:
					ans = max(ans, max(A[i - k:i + 1]) * (k + 1) + dp(i - k - 1))
			return ans
		return dp(len(A) - 1)


class Solution:

	def maxSumAfterPartitioning(self, arr, k):
		if not arr:
			return 0
		dp = [0] * len(arr)
		for i in range(len(arr)):
			for j in range(k):
				if i - j < 0:
					continue
				max_val = max(arr[i - j:i + 1])
				if i - j == 0:
					total = max_val * (j + 1)
				else:
					total = dp[i - j - 1] + max_val * (j + 1)
				dp[i] = max(dp[i], total)
		return dp[-1]


class Solution:

	def maxSumAfterPartitioning(self, A: List[int], K: int) -> int:
		N = len(A)
		dp = [0] * N
		dp[0] = A[0]
		for i in range(1, N):
			for j in range(0, min(K, i + 1)):
				current_subarray = A[i - j:i + 1]
				if i >= K:
					dp[i] = max(dp[i], dp[i - j - 1] + max(current_subarray) * (j + 1))
				else:
					dp[i] = max(dp[i], max(current_subarray) * (j + 1))
		return dp[-1]
		(0, 1)
		(1, 1)


class Solution:

	def maxSumAfterPartitioning(self, A: List[int], K: int) -> int:
		n = len(A)
		dp = [0] * n
		for i in range(n):
			if i == 0:
				dp[i] = A[0]
			else:
				for k in range(1, K + 1):
					if i - k + 1 < 0:
						break
					elif i - k + 1 == 0:
						dp[i] = max(dp[i], max(A[i - k + 1:i + 1]) * k)
					else:
						dp[i] = max(dp[i], dp[i - k] + max(A[i - k + 1:i + 1]) * k)
		return dp[n - 1]


class Solution:

	def maxSumAfterPartitioning(self, A: List[int], K: int) -> int:
		n = len(A)
		dp = [0] * (n + 1)
		for i in range(n):
			for k in range(1, K + 1):
				if i - k > -2:
					dp[i + 1] = max(dp[i + 1], max(A[max(0, i - k + 1):i + 1]) * k + dp[i + 1 - k])
		return dp[n]


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		dp = [0] * len(arr)
		for i in range(len(arr)):
			print(arr[i])
			max_so_far = arr[i] + (dp[i - 1] if i > 0 else 0)
			for j in range(1, min(i + 1, k)):
				max_in_range = max(arr[i - j:i + 1])
				max_so_far = max(max_so_far, dp[i - j - 1] + max_in_range * (j + 1))
			dp[i] = max_so_far
		return dp[len(arr) - 1]


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		dp = [0] * len(arr)
		dp[0] = arr[0]
		for (i, val) in enumerate(arr):
			if i == 0:
				continue
			cur_max = val + dp[i - 1]
			cur_k = 1
			while i - cur_k >= 0 and cur_k < k:
				temp = arr[i - cur_k:i + 1]
				m = max(temp)
				if i - cur_k == 0:
					cur_max = max(cur_max, m * (cur_k + 1))
				else:
					cur_max = max(cur_max, m * (cur_k + 1) + dp[i - cur_k - 1])
				cur_k += 1
			dp[i] = cur_max
		return dp[-1]


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		DP = [0 for _ in range(len(arr) + 1)]
		for i in range(1, len(DP)):
			loc = max(i - k, 0)
			run_max = 0
			for j in range(loc, i):
				run_max = max(run_max, DP[j] + max(arr[j:i]) * (i - j))
			DP[i] = run_max
		return DP[-1]


class Solution:

	def maxSumAfterPartitioning(self, A: List[int], K: int) -> int:
		if not K or K == 1:
			return sum(A)
		if not A:
			return 0
		dp = [0] * len(A)
		for (index, num) in enumerate(A):
			possible = []
			for group in range(K):
				if index - group >= 0:
					if index - group - 1 >= 0:
						previous = dp[index - group - 1]
					else:
						previous = 0
					possible.append(previous + max(A[index - group:index + 1]) * (group + 1))
			dp[index] = max(possible)
		return dp[-1]


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		self.arr = arr
		self.dp = {}

		def helper(i, j, k):
			if j <= i:
				return 0
			if (i, j) in self.dp:
				return self.dp[i, j]
			maxsum = -sys.maxsize
			for length in range(1, k + 1):
				if i + length <= j:
					currsum = max(self.arr[i:i + length]) * length + helper(i + length, j, k)
					maxsum = max(maxsum, currsum)
			self.dp[i, j] = maxsum
			return self.dp[i, j]
		return helper(0, len(arr), k)


class Solution:

	def maxSumAfterPartitioning(self, A: List[int], K: int) -> int:
		dp = [0 for _ in A]
		for (i, num) in enumerate(A):
			for j in range(K):
				ans = 0
				if i - j >= 0:
					ans = ans + (j + 1) * max(A[i - j:i + 1])
				if i - j - 1 >= 0:
					ans = ans + dp[i - j - 1]
				dp[i] = max(dp[i], ans)
		return dp[-1]


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		d = [0] * (1 + len(arr) + k)
		for ind in range(1, len(arr) + 1):
			for sub_arr_st in range(min(ind, k)):
				if arr[ind - sub_arr_st - 1] > arr[ind - 1]:
					break
				for sub_arr_len in range(sub_arr_st + 1, k + 1):
					ind_x = ind - sub_arr_st + sub_arr_len - 1
					d[ind_x] = max(d[ind_x], d[ind - sub_arr_st - 1] + arr[ind - 1] * sub_arr_len)
		return d[len(arr)]


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		dp = [0] * (len(arr) + 1)
		dp[0] = arr[0]
		for i in range(1, len(arr)):
			ans = -100000
			for z in range(max(0, i - k + 1), i + 1):
				ans = max(dp[z - 1] + max(arr[z:i + 1]) * (i - z + 1), ans)
			dp[i] = ans
		return dp[-2]


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		memo = {}
		l_arr = len(arr)
		for chain_len in range(0, k, 1):
			for i in range(0, l_arr - chain_len, 1):
				memo[i, i + chain_len] = max(arr[i:i + chain_len + 1]) * (chain_len + 1)
		res_memo = {}

		def dfs(idx):
			if idx == l_arr:
				return 0
			if idx in res_memo:
				return res_memo[idx]
			res_memo[idx] = max((memo[idx, idx + j] + dfs(idx + j + 1) for j in range(k) if idx + j < l_arr))
			return res_memo[idx]
		return dfs(0)


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		res = [0] * (len(arr) + 1)
		for i in range(1, len(arr) + 1):
			for j in range(1, k + 1):
				if i - j > -1:
					res[i] = max(res[i], res[i - j] + max(arr[i - j:i]) * j)
		return res[-1]


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		ar_len = len(arr)
		dp = [0] * (ar_len + 1)
		for i in range(1, len(dp)):
			tmp_max = []
			for j in range(1, k + 1):
				if i - j >= 0:
					max_t = dp[i - j] + max(arr[i - j:i]) * j
					tmp_max.append(max_t)
			dp[i] = max(tmp_max)
		return dp[-1]


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		n = len(arr)
		if k == n:
			return n * max(arr)
		dp = collections.deque([0] * k)
		for i in range(n - 1, -1, -1):
			m = 0
			result = 0
			for j in range(min(k, n - i)):
				m = max(m, arr[i + j])
				result = max(result, m * (j + 1) + dp[j])
			dp.appendleft(result)
			dp.pop()
		return dp[0]


class Solution:

	def maxSumAfterPartitioning(self, nums: List[int], k: int) -> int:
		dp = [0] * len(nums)
		for i in range(len(nums)):
			for windowSize in range(1, k + 1):
				startIndex = i - windowSize + 1
				if startIndex < 0:
					break
				maxVal = max(nums[startIndex:i + 1])
				currVal = windowSize * maxVal
				currSum = currVal
				if startIndex > 0:
					currSum += dp[startIndex - 1]
				dp[i] = max(dp[i], currSum)
		return dp[-1]


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		n = len(arr)
		memo = {n: 0}

		def helper(arr, i):
			if i in memo:
				return memo[i]
			res = 0
			for j in range(i, min(n, i + k)):
				res = max(res, helper(arr, j + 1) + max(arr[i:j + 1]) * (j + 1 - i))
			memo[i] = res
			return res
		return helper(arr, 0)


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		result = [0] * len(arr)
		result[0] = arr[0]
		max_val = result[0]
		for i in range(1, k):
			max_val = max(max_val, arr[i])
			result[i] = max_val * (i + 1)
		for i in range(k, len(arr)):
			max_val = arr[i]
			for j in range(1, k + 1):
				max_val = max(max_val, arr[i - j + 1])
				result[i] = max(result[i], result[i - j] + max_val * j)
		return result[-1]


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		A = arr
		K = k
		n = len(A)
		dp = [0] * n
		curMax = 0
		for i in range(n):
			if i < K:
				curMax = max(curMax, A[i])
				dp[i] = curMax * (i + 1)
			else:
				curMax = 0
				for j in range(1, K + 1):
					curMax = max(A[i - j + 1], curMax)
					dp[i] = max(dp[i], dp[i - j] + curMax * j)
		return dp[n - 1]


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		n = len(arr)
		dp = [0] * (n + 1)
		for i in range(1, n + 1):
			max_val = float('-inf')
			for j in range(1, min(i, k) + 1):
				max_val = max(max_val, arr[i - j])
				dp[i] = max(dp[i], dp[i - j] + j * max_val)
		return dp[n]


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		f = [0]
		for end in range(1, len(arr) + 1):
			_local = 0
			_cur_max = 0
			for start in range(end - 1, max(0, end - k) - 1, -1):
				_cur_max = max(_cur_max, arr[start])
				_local = max(_cur_max * (end - start) + f[start], _local)
			f.append(_local)
		return f[-1]


class Solution:

	def maxSumAfterPartitioning(self, arr, k):
		n = len(arr)
		dp = [0] * (n + 1)
		for i in range(n):
			cmax = 0
			for j in range(1, min(k, i + 1) + 1):
				cmax = max(cmax, arr[i - j + 1])
				dp[i] = max(dp[i], dp[i - j] + cmax * j)
		return dp[-2]


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		best_sums = [0] * (len(arr) + 1)
		for i in range(len(arr)):
			curr_max = 0
			for j in range(1, min(k + 1, i + 2)):
				curr_max = max(curr_max, arr[i - j + 1])
				best_sums[i] = max(best_sums[i], best_sums[i - j] + j * curr_max)
		return best_sums[-2]


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		DP = [0] * len(arr)
		N = len(arr)
		DP[0] = arr[0]
		m = arr[0]
		for i in range(k):
			m = max(m, arr[i])
			DP[i] = m * (i + 1)
		for i in range(k, N):
			mm = arr[i]
			DP[i] = DP[i - 1] + arr[i]
			for j in range(1, k):
				mm = max(mm, arr[i - j])
				DP[i] = max(DP[i], DP[i - j - 1] + mm * (j + 1))
		return DP[-1]


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		n = len(arr)
		dp = [0] * (n + 1)
		for i in range(n):
			cur = 0
			for j in range(1, min(k, i + 1) + 1):
				cur = max(cur, arr[i - j + 1])
				dp[i] = max(dp[i], cur * j + dp[i - j])
		return dp[-2]


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:

		@lru_cache(None)
		def dp(idx):
			if idx < 0:
				return 0
			left = max(idx - k + 1, 0)
			curr_max = arr[idx]
			res = arr[idx]
			for i in range(idx, left - 1, -1):
				curr_max = max(curr_max, arr[i])
				res = max(res, dp(i - 1) + (idx - i + 1) * curr_max)
			return res
		return dp(len(arr) - 1)


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		N = len(arr)
		dp = [0] * (N + 1)
		for i in range(N):
			cur_max = 0
			for j in range(1, min(k, i + 1) + 1):
				cur_max = max(cur_max, arr[i - j + 1])
				dp[i] = max(dp[i], dp[i - j] + cur_max * j)
		return dp[N - 1]


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		N = len(arr)
		dp = [0 for _ in range(N)]
		dp[0] = arr[0]
		max_so_far = arr[0]
		for i in range(1, k):
			max_so_far = max(max_so_far, arr[i])
			dp[i] = (i + 1) * max_so_far
		for i in range(k, N):
			max_so_far = -sys.maxsize
			for j in range(i, i - k, -1):
				max_so_far = max(max_so_far, arr[j])
				dp[i] = max(dp[i], dp[j - 1] + (i - j + 1) * max_so_far)
		return dp[N - 1]


class Solution:

	def maxSumAfterPartitioning(self, A: List[int], K: int) -> int:
		n = len(A)
		dp = [0] * (n + 1)
		for i in range(1, n + 1):
			j = i - 1
			mx = -float('inf')
			while i - j <= K and j >= 0:
				mx = max(mx, A[j])
				dp[i] = max(dp[i], dp[j] + mx * (i - j))
				j -= 1
		return dp[n]


class Solution:

	def maxSumAfterPartitioning(self, A: List[int], K: int) -> int:
		N = len(A)
		DP = [0] * (N + 1)
		for i in range(N):
			curMax = 0
			for k in range(1, min(K, i + 1) + 1):
				curMax = max(curMax, A[i - k + 1])
				DP[i] = max(DP[i], DP[i - k] + curMax * k)
		return DP[N - 1]


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		n = len(arr)
		dp = [0] * (n + 1)
		for i in range(n):
			cur_max = 0
			for d in range(1, min(k, i + 1) + 1):
				cur_max = max(cur_max, arr[i - d + 1])
				dp[i + 1] = max(dp[i + 1], dp[i + 1 - d] + cur_max * d)
		return dp[-1]


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		dp = [0]
		for i in range(1, len(arr) + 1):
			max_arr_num = 0
			now_max = 0
			for j in range(1, k + 1):
				idx = i - j
				if idx >= 0:
					max_arr_num = max(max_arr_num, arr[idx])
					now_max = max(now_max, dp[idx] + max_arr_num * j)
			dp.append(now_max)
		return dp[-1]


class Solution:

	def maxSumAfterPartitioning(self, A: List[int], K: int) -> int:
		n = len(A)
		dp = [0] * (n + 1)
		for i in range(1, n + 1):
			m = float('-inf')
			for j in range(1, K + 1):
				if i - j >= 0:
					m = max(A[i - j], m)
					dp[i] = max(dp[i], dp[i - j] + m * j)
		return dp[-1]


class Solution:

	def maxSumAfterPartitioning(self, A: List[int], K: int) -> int:
		n = len(A)
		dp = [0] * (n + 1)
		res = 0
		for i in range(n):
			max_val = 0
			for k in range(1, min(K, i + 1) + 1):
				max_val = max(max_val, A[i - k + 1])
				dp[i] = max(dp[i], dp[i - k] + max_val * k)
		return dp[n - 1]


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		d = [0] * (len(arr) + 1)
		for i in range(len(arr)):
			cur_max = 0
			for kk in range(1, k + 1):
				if i - kk + 1 < 0:
					continue
				cur_max = max(cur_max, arr[i - kk + 1])
				d[i] = max(d[i], d[i - kk] + cur_max * kk)
		print(d)
		return d[-2]


class Solution:

	def maxSumAfterPartitioning(self, A: List[int], K: int) -> int:
		N = len(A)
		dp = [0] * (N + 1)
		for i in range(N):
			curr_max = 0
			for j in range(1, K + 1):
				if i + j > N:
					break
				curr_max = max(curr_max, A[i + j - 1])
				dp[i + j] = max(dp[i + j], dp[i] + curr_max * j)
		return dp[-1]


class Solution:

	def maxSumAfterPartitioning(self, A: List[int], K: int) -> int:
		dp = [float('-inf') for i in range(len(A) + 1)]
		dp[-1] = 0
		dp[-2] = A[-1]
		for j in reversed(list(range(len(A) - 1))):
			cur_max = float('-inf')
			for k in range(K):
				if j + k == len(A):
					break
				cur_max = max(cur_max, A[j + k])
				dp[j] = max(dp[j], (k + 1) * cur_max + dp[j + k + 1])
		return dp[0]


class Solution:

	def maxSumAfterPartitioning(self, A: List[int], k: int) -> int:
		N = len(A)
		dp = [0] * (N + 1)
		for i in range(N):
			mx = 0
			for j in range(i, max(-1, i - k), -1):
				mx = max(mx, A[j])
				p = dp[i + 1]
				dp[i + 1] = max(dp[i + 1], dp[j] + mx * (i - j + 1))
		return dp[N]


class Solution:

	def maxSumAfterPartitioning(self, A, K):
		n = len(A)
		record = [0] * (n + 1)
		for i in range(n):
			curMax = 0
			for k in range(1, min(K, i + 1) + 1):
				curMax = max(curMax, A[i - k + 1])
				record[i + 1] = max(record[i + 1], record[i - k + 1] + curMax * k)
		return record[n]


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		n = len(arr)
		dp = [0] * (n + 1)
		for i in range(1, n + 1):
			m = -float('inf')
			d = 0
			while d < k and i - d > 0:
				m = max(m, arr[i - 1 - d])
				dp[i] = max(dp[i], dp[i - d - 1] + m * (d + 1))
				d += 1
		return dp[n]


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		N = len(arr)
		memo = {}

		def maxSum2(i):
			if i >= N:
				return 0
			if i in memo:
				return memo[i]
			val = -1
			sol = -1
			for j in range(1, min(k + 1, N - i + 1)):
				val = max(val, arr[i + j - 1])
				sol = max(sol, val * j + maxSum2(i + j))
			memo[i] = sol
			return sol
		return maxSum2(0)


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		n = len(arr)
		dp = [0] * (n + 1)
		for i in range(n):
			currMax = arr[i]
			size = 1
			while size <= k and i - size + 1 >= 0:
				currMax = max(currMax, arr[i - size + 1])
				dp[i + 1] = max(dp[i + 1], dp[i + 1 - size] + currMax * size)
				size += 1
		return dp[n]


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		memo = {}

		def max_sum(idx):
			if idx >= len(arr):
				return 0
			if idx in memo:
				return memo[idx]
			subarr_max = float('-inf')
			options = []
			for end in range(idx, min(idx + k, len(arr))):
				subarr_max = max(subarr_max, arr[end])
				subarr_sum = subarr_max * (end - idx + 1)
				options.append(subarr_sum + max_sum(end + 1))
			memo[idx] = max(options)
			return memo[idx]
		return max_sum(0)


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		n = len(arr)
		memo = {}

		def dfs(i):
			if i >= n:
				return 0
			if i in memo:
				return memo[i]
			res = float('-inf')
			cur_max = float('-inf')
			for j in range(i, min(i + k, n)):
				cur_max = max(cur_max, arr[j])
				res = max(res, cur_max * (j - i + 1) + dfs(j + 1))
			memo[i] = res
			return res
		return dfs(0)


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		N = len(arr)
		maxSoFar = [0] * N
		for i in range(N):
			curMax = 0
			for prev in range(1, min(i + 1, k) + 1):
				curMax = max(curMax, arr[i - prev + 1])
				lastPartition = maxSoFar[i - prev] if i >= prev else 0
				maxSoFar[i] = max(maxSoFar[i], lastPartition + curMax * prev)
		return maxSoFar[-1]


class Solution:

	def maxSumAfterPartitioning(self, A: List[int], K: int) -> int:

		def dfs(start, memo):
			if start in memo:
				return memo[start]
			N = len(A)
			if start >= N:
				return 0
			maxSum = 0
			maxEle = 0
			for i in range(start, min(N, start + K)):
				maxEle = max(maxEle, A[i])
				maxSum = max(maxSum, maxEle * (i - start + 1) + dfs(i + 1, memo))
			memo[start] = maxSum
			return maxSum
		return dfs(0, {})


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		if arr == [] or k == 0:
			return 0
		if len(arr) == 1 and k == 1:
			return arr[0]
		l = len(arr)
		res = [0] * l
		for i in range(0, l):
			sub_max = 0
			for j in range(k):
				if j <= i:
					sub_max = max(sub_max, arr[i - j])
					res[i] = max(res[i], sub_max * (j + 1) + (res[i - j - 1] if i - j - 1 >= 0 else 0))
		return res[l - 1]


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		dp = len(arr) * [0]
		dp[0] = arr[0]
		for i in range(1, len(arr)):
			sub_max = 0
			for j in range(k):
				if j <= i:
					sub_max = max(sub_max, arr[i - j])
					dp[i] = max(dp[i], sub_max * (j + 1) + (dp[i - j - 1] if i - j - 1 >= 0 else 0))
		return dp[len(arr) - 1]


class Solution:

	def maxSumAfterPartitioning(self, A: List[int], K: int) -> int:
		dp_sum = [0 for _ in range(len(A) + 1)]
		for i in range(len(A)):
			seg_max = A[i]
			for j in range(1, K + 1):
				if i - j + 1 < 0:
					break
				seg_max = max(seg_max, A[i - j + 1])
				tmp = dp_sum[i - j + 1] + seg_max * j
				dp_sum[i + 1] = max(dp_sum[i + 1], tmp)
		return dp_sum[-1]


class Solution:

	def maxSumAfterPartitioning(self, A: List[int], k: int) -> int:
		L = len(A)
		vis = {}

		def pos(n):
			if n >= L:
				return 0
			if n in vis:
				return vis[n]
			currmax = A[n]
			ans = A[n] + pos(n + 1)
			for i in range(1, k):
				n1 = n + i
				if n1 >= L:
					break
				currmax = max(currmax, A[n1])
				ans = max(ans, currmax * (i + 1) + pos(n1 + 1))
			vis[n] = ans
			return vis[n]
		return pos(0)


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		if not arr:
			return 0
		n = len(arr)
		dp = [0 for _ in range(n)]
		for i in range(n):
			temp_max = arr[i]
			j = 1
			while i - j + 1 >= 0 and j <= k:
				temp_max = max(temp_max, arr[i - j + 1])
				if i >= j:
					dp[i] = max(dp[i], dp[i - j] + temp_max * j)
				else:
					dp[i] = max(dp[i], temp_max * j)
				j += 1
		return dp[-1]


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		n = len(arr)
		dp = [-1 for _ in range(n)]

		def max_sum(i):
			if i >= n:
				return 0
			if i == n - 1:
				return arr[i]
			if dp[i] != -1:
				return dp[i]
			ans = float('-inf')
			maxi = arr[i]
			for p in range(1, k + 1):
				v = maxi * p + max_sum(i + p)
				ans = max(v, ans)
				if i + p < n:
					maxi = max(maxi, arr[i + p])
				else:
					break
			dp[i] = ans
			return ans
		v = max_sum(0)
		return v


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		dp = {}

		def help(arr, k, start=0):
			if start == len(arr):
				return 0
			if start in dp:
				return dp[start]
			dp[start] = -float('inf')
			maxval = arr[start]
			for i in range(start, min(start + k, len(arr))):
				maxval = max(maxval, arr[i])
				dp[start] = max(dp[start], maxval * (i - start + 1) + help(arr, k, i + 1))
			return dp[start]
		return help(arr, k)


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		N = len(arr)
		K = k
		dp = [0] * (N + 1)
		for i in range(N):
			curMax = 0
			for k in range(1, min(K, i + 1) + 1):
				curMax = max(curMax, arr[i - k + 1])
				dp[i] = max(dp[i], dp[i - k] + curMax * k)
		return dp[N - 1]


class Solution:

	def maxSumAfterPartitioning(self, A: List[int], K: int) -> int:
		memo = {}

		def getMax(arr, k, idx):
			if idx == len(arr):
				return 0
			if idx in memo:
				return memo[idx]
			(maxSum, maxInSub) = (0, 0)
			for i in range(idx, min(idx + K, len(arr))):
				maxInSub = max(maxInSub, arr[i])
				maxSum = max(maxSum, maxInSub * (i - idx + 1) + getMax(arr, k, i + 1))
			memo[idx] = maxSum
			return maxSum
		return getMax(A, K, 0)


class Solution:

	def maxSumAfterPartitioning(self, A: List[int], K: int) -> int:
		dp = [0] * len(A)
		dp[0] = A[0]
		partition_max = A[0]
		for x in range(1, K):
			partition_max = max(partition_max, A[x])
			dp[x] = (x + 1) * partition_max
		print(dp)
		for i in range(K, len(dp)):
			partition_max = float('-inf')
			for j in range(K):
				partition_max = max(partition_max, A[i - j])
				dp[i] = max(dp[i], dp[i - j - 1] + (j + 1) * partition_max)
		print(dp)
		return dp[-1]


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		n = len(arr)
		dp = [0] * (n + 1)
		dp[1] = arr[0]
		for i in range(2, n + 1):
			dp[i] = dp[i - 1] + arr[i - 1]
			cur_max = arr[i - 1]
			for j in range(1, k + 1):
				cur_max = max(cur_max, arr[i - j])
				if i - j >= 0:
					dp[i] = max(dp[i], dp[i - j] + j * cur_max)
		return dp[n]


class Solution:

	def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:
		dp = [0] * (len(arr) + 1)
		for i in range(1, len(arr) + 1):
			dp[i] = float('-inf')
			curr_max = arr[i - 1]
			for p in range(1, k + 1):
				if p > i:
					break
				curr_max = max(curr_max, arr[i - p])
				dp[i] = max(dp[i], p * curr_max + dp[i - p])
		return dp[-1]


class Solution:

	def maxSumAfterPartitioning(self, A: List[int], K: int) -> int:
		self.max_sum = float('-inf')
		self.dic = {}

		def getSum(i):
			if i >= len(A):
				return 0
			if i in self.dic:
				return self.dic[i]
			cur_max = A[i]
			ret = cur_max + getSum(i + 1)
			for j in range(1, K):
				if i + j < len(A):
					if A[i + j] > cur_max:
						cur_max = A[i + j]
					ret = max(cur_max * (j + 1) + getSum(i + j + 1), ret)
				else:
					break
			self.dic[i] = ret
			return ret
		return getSum(0)


class Solution:

	@lru_cache(None)
	def rec(self, idx):
		if idx > len(self.A):
			return 0
		max_yet = 0
		max_score = 0
		for i in range(self.K):
			if idx + i >= len(self.A):
				break
			max_yet = max(max_yet, self.A[idx + i])
			max_score = max(max_score, max_yet * (i + 1) + self.rec(idx + i + 1))
		return max_score

	def maxSumAfterPartitioning(self, A: List[int], K: int) -> int:
		self.A = A
		self.K = K
		return self.rec(0)


class Solution:

	def maxSumAfterPartitioning(self, A: List[int], K: int) -> int:

		def DPHelper(index):
			if index >= len(A):
				return 0
			if memo.get(index) is not None:
				return memo[index]
			res = 0
			curr_max = 0
			for i in range(index, min(index + K, len(A))):
				curr_max = max(curr_max, A[i])
				res = max(res, curr_max * (i - index + 1) + DPHelper(i + 1))
			memo[index] = res
			return res
		memo = {}
		return DPHelper(0)