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

```python

USER_FILENAME = "<user_code>"
import sys, json, builtins
import math, heapq, bisect, itertools, functools, collections, statistics, fractions, decimal, operator, array, random, re, string, os, pathlib
try:
    import resource as _res
except Exception:
    _res = None
from collections import Counter, defaultdict, deque, OrderedDict
from functools import lru_cache, reduce
from itertools import accumulate, permutations, combinations, product
from math import gcd, ceil, floor, sqrt
from decimal import Decimal
try:
    from typing import *
except Exception:
    pass
try:
    sys.setrecursionlimit(1_000_000)
except Exception:
    pass
try:
    import site as _site
    try:
        _sys_sites = _site.getsitepackages() if hasattr(_site, "getsitepackages") else []
        for _p in _sys_sites:
            if _p and _p not in sys.path:
                sys.path.append(_p)
    except Exception:
        pass
    try:
        _usp = _site.getusersitepackages()
        if isinstance(_usp, str):
            _usp = [_usp]
        for _p in _usp:
            if _p and _p not in sys.path:
                sys.path.append(_p)
    except Exception:
        pass
except Exception:
    pass

def _disabled_input(*args, **kwargs):
    raise RuntimeError("input() is disabled in callable mode")
builtins.input = _disabled_input

ns = {"__name__": "__judge__"}  # prevent if __name__ == "__main__" blocks
for _k, _v in list(globals().items()):
    if not _k.startswith("_"):
        ns[_k] = _v
_CODE = "class Solution:\n    def getMax(self, arr, k, idx, cache):\n        # I am partitioning 1 element\n        if idx == len(arr) - 1:\n            return arr[idx]\n        # I need to get max sum after partitioning\n        maxSum = 0\n        \n        # [1,2,9,30]\n        # I need to try partitioning from 1 -> K elements\n        for numInPartition in range(1, k + 1):\n            # I cannot choose elements if I would be choosing too many elements\n            if idx + numInPartition > len(arr):\n                break\n            \n            startOfRecursiveIndex = idx + numInPartition\n            # select max value in first 'k' elements\n            maxVal = max(arr[idx:startOfRecursiveIndex])\n            \n            # generates sum for this subset\n            partSum = maxVal * numInPartition\n            \n            # I need to get the partition sum for the rest of the elements\n            \n            rest = cache[startOfRecursiveIndex] if startOfRecursiveIndex in cache else self.getMax(arr, k, startOfRecursiveIndex, cache)\n            \n            cache[startOfRecursiveIndex] = rest\n            \n            maxSum = max(maxSum, partSum + rest)\n        return maxSum\n            \n    def maxSumAfterPartitioning(self, A: List[int], K: int) -> int:\n        '''\n        Input:\n            A: int[] - Array of numbers\n            K: int - Max length allowable for subarray\n            \n        Output:\n            Return largest sum after partitioning.\n        \n        Steps:\n            1. Make partition\n            2. Change all elements to max value\n            \n           v            \n        [1,2,9,30], 2\n        \n        {1,2}{9,30} => {2,2}{30,30} => 4 + 60 = 64\n        {1}{2,9}{30} => {1}{9,9}{30} => 1 + 18 + 30 = 49\n        \n        \n        Example:\n            [1]\n            {1} = 1\n            \n            [1,2]\n            {1}{2} => 1+2 = 3\n            {1,2} => {2,2} => 4\n                 v\n            [1,2,9]\n            {1,2}{9} => {2,2} + {9} => 4 + 9 = 13\n            {1}{2,9} => {1},{9,9} = 1 + 18 = 19\n            {1,2,9}\n            \n            {1}{2,9}{30} => {1}{9,9}{30} => 19 + 30 => 49\n            {1,2}{9,30} => {2,2}{30,30} => 4 + 60 => 64\n            \n                 v\n            [1,2,9,30]\n            {1,2,9} {30} => 30 + getMaxSum([1,2,9]) => 30 + 19 => 49\n            {1,2} {9,30} => 60 + getMaxSum([1,2]) => 60 + 4 = 64\n            \n            {1}{2,9,30}\n            {1,2}{9,30}\n            {1,2,9}{30}\n        \n        All about subproblems!!\n        \n        Approach:\n            - Let's look into sliding window.\n            - We can frame this problem in terms of choices.\n                When considering an element, we can place it into its own partition OR with the preceeding partition.\n                Choice:\n                    K-way choice\n                    Do I group my current element in the last [1,k] elements?\n                Constraint:\n                    Subset needs to be contiguous?\n                Goal:\n                    Maximum sum\n                \n            1. When placing\n        Approach:\n            [0,0,0,0]\n            maxSum[i] = max(maxSum[i-1] + A[i] * 1, maxSum[i-2] + max(last two elements) * 2) and so on\n        '''\n        \n        cache = {}\n        return self.getMax(A, K, 0, cache)\n       \n    \n    \n    \n    \n"
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

_target = None
_fn_name = 'maxSumAfterPartitioning'
if _fn_name:
    if "Solution" in ns:
        try:
            _obj = ns["Solution"]()
            if hasattr(_obj, _fn_name):
                _target = getattr(_obj, _fn_name)
        except Exception:
            _target = None
    if _target is None and _fn_name in ns and callable(ns[_fn_name]):
        _target = ns[_fn_name]
# Fallback: if there's exactly one user-defined function, use it
if _target is None:
    _cands = [v for k, v in ns.items() if callable(v) and getattr(v, "__module__", "") in ("__judge__", "__main__")]
    _user_funcs = [c for c in _cands if getattr(c, "__name__", "").startswith("_") is False]
    if len(_user_funcs) == 1:
        _target = _user_funcs[0]
if _target is None:
    raise RuntimeError(f"Could not resolve callable '{_fn_name}'")

_raw = sys.stdin.read().strip()
if _raw:
    _parsed = json.loads(_raw)
    if isinstance(_parsed, dict):
        _kwargs = _parsed.get("kwargs", {})
        _args = _parsed.get("args", [])
    else:
        _args, _kwargs = _parsed, {}
else:
    _args, _kwargs = [], {}
if not isinstance(_args, (list, tuple)):
    _args = [_args]
_res = _target(*_args, **_kwargs)
try:
    sys.stdout.write(repr(_res) + "\n")
except Exception:
    pass

```

Measured memory_bytes: 7726