<p>Given a 2D character matrix <code>grid</code>, where <code>grid[i][j]</code> is either <code>&#39;X&#39;</code>, <code>&#39;Y&#39;</code>, or <code>&#39;.&#39;</code>, return the number of <span data-keyword="submatrix">submatrices</span> that contain:</p>

<ul>
	<li><code>grid[0][0]</code></li>
	<li>an <strong>equal</strong> frequency of <code>&#39;X&#39;</code> and <code>&#39;Y&#39;</code>.</li>
	<li><strong>at least</strong> one <code>&#39;X&#39;</code>.</li>
</ul>

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

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">grid = [[&quot;X&quot;,&quot;Y&quot;,&quot;.&quot;],[&quot;Y&quot;,&quot;.&quot;,&quot;.&quot;]]</span></p>

<p><strong>Output:</strong> <span class="example-io">3</span></p>

<p><strong>Explanation:</strong></p>

<p><strong><img alt="" src="https://assets.leetcode.com/uploads/2024/06/07/examplems.png" style="padding: 10px; background: rgb(255, 255, 255); border-radius: 0.5rem; width: 175px; height: 350px;" /></strong></p>
</div>

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

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">grid = [[&quot;X&quot;,&quot;X&quot;],[&quot;X&quot;,&quot;Y&quot;]]</span></p>

<p><strong>Output:</strong> <span class="example-io">0</span></p>

<p><strong>Explanation:</strong></p>

<p>No submatrix has an equal frequency of <code>&#39;X&#39;</code> and <code>&#39;Y&#39;</code>.</p>
</div>

<p><strong class="example">Example 3:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">grid = [[&quot;.&quot;,&quot;.&quot;],[&quot;.&quot;,&quot;.&quot;]]</span></p>

<p><strong>Output:</strong> <span class="example-io">0</span></p>

<p><strong>Explanation:</strong></p>

<p>No submatrix has at least one <code>&#39;X&#39;</code>.</p>
</div>

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

<ul>
	<li><code>1 &lt;= grid.length, grid[i].length &lt;= 1000</code></li>
	<li><code>grid[i][j]</code> is either <code>&#39;X&#39;</code>, <code>&#39;Y&#39;</code>, or <code>&#39;.&#39;</code>.</li>
</ul>

class Solution:
    def numberOfSubmatrices(self, grid: List[List[str]]) -> int:
        ROWS = len(grid)
        COLS = len(grid[0])
        prefix_x = collections.defaultdict(lambda: collections.defaultdict(int))
        prefix_y = collections.defaultdict(lambda: collections.defaultdict(int))
        
        for r in range(ROWS):
            prefix_row_x = 0
            prefix_row_y = 0
            for c in range(COLS):
                prefix_row_x += 1 if grid[r][c] == "X" else 0
                prefix_row_y += 1 if grid[r][c] == "Y" else 0
                
                above_x = prefix_x[r-1][c]
                above_y = prefix_y[r-1][c]
                
                prefix_x[r][c] = above_x + prefix_row_x
                prefix_y[r][c] = above_y + prefix_row_y
        
        count = 0
        for r in range(ROWS):
            for c in range(COLS):
                if prefix_x[r][c] == prefix_y[r][c] and prefix_x[r][c] > 0:
                    count += 1
        return count
    
# Time: O(N^2)
# Space: O(N^2)
                
        
        
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