<p>You are given a 2D matrix <code>grid</code> of size <code>n x n</code>. Initially, all cells of the grid are colored white. In one operation, you can select any cell of indices <code>(i, j)</code>, and color black all the cells of the <code>j<sup>th</sup></code> column starting from the top row down to the <code>i<sup>th</sup></code> row.</p>

<p>The grid score is the sum of all <code>grid[i][j]</code> such that cell <code>(i, j)</code> is white and it has a horizontally adjacent black cell.</p>

<p>Return the <strong>maximum</strong> score that can be achieved after some number of operations.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">grid = [[0,0,0,0,0],[0,0,3,0,0],[0,1,0,0,0],[5,0,0,3,0],[0,0,0,0,2]]</span></p>

<p><strong>Output:</strong> <span class="example-io">11</span></p>

<p><strong>Explanation:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2024/05/11/one.png" style="width: 300px; height: 200px;" />
<p>In the first operation, we color all cells in column 1 down to row 3, and in the second operation, we color all cells in column 4 down to the last row. The score of the resulting grid is <code>grid[3][0] + grid[1][2] + grid[3][3]</code> which is equal to 11.</p>
</div>

<p><strong class="example">Example 2:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">grid = [[10,9,0,0,15],[7,1,0,8,0],[5,20,0,11,0],[0,0,0,1,2],[8,12,1,10,3]]</span></p>

<p><strong>Output:</strong> <span class="example-io">94</span></p>

<p><strong>Explanation:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2024/05/11/two-1.png" style="width: 300px; height: 200px;" />
<p>We perform operations on 1, 2, and 3 down to rows 1, 4, and 0, respectively. The score of the resulting grid is <code>grid[0][0] + grid[1][0] + grid[2][1] + grid[4][1] + grid[1][3] + grid[2][3] + grid[3][3] + grid[4][3] + grid[0][4]</code> which is equal to 94.</p>
</div>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;=&nbsp;n == grid.length &lt;= 100</code></li>
	<li><code>n == grid[i].length</code></li>
	<li><code>0 &lt;= grid[i][j] &lt;= 10<sup>9</sup></code></li>
</ul>

class Solution:
    def maximumScore(self, grid: List[List[int]]) -> int:
        n = len(grid)
        if n == 1:
            return 0
        ans = 0
        cumscore = [[0 for i in range(n + 1)] for j in range(n)]
        for j in range(n):
            for i in range(n):
                cumscore[j][i+1] += cumscore[j][i] + grid[i][j]
        
        pi = [0] * (n+1)
        pe = [0] * (n+1)
        for j in range(1,n):
            scoreinclu = [0] * (n+1)
            scoreexclu = [0] * (n+1)
            for i in range(n+1):
                for k in range(n+1):
                    scoreinclu[i] = max(scoreinclu[i], pi[k] + max(0, cumscore[j][k] - cumscore[j][i]), pe[k] + max(0, cumscore[j-1][i] - cumscore[j-1][k]))
                    scoreexclu[i] = max(scoreexclu[i], pi[k] , pe[k] + max(0, cumscore[j-1][i] - cumscore[j-1][k]))
            pi = scoreinclu
            pe = scoreexclu
        
        return max(pi)
                    


                    
                    
                
18100