<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:
        m = len(grid)
        n = len(grid[0])
        dp = [[(0, 0)] * n for _ in range(m)]
        result = 0
        # Populate first row
        for i in range(n):
            if i == 0:
                if grid[0][0] == "X":
                   dp[0][0] = (1, 0)
                elif grid[0][0] == "Y":
                    dp[0][0] = (0, 1)
            else:
                x, y = dp[0][i - 1]
                if grid[0][i] == "X":
                    x += 1
                elif grid[0][i] == "Y":
                    y += 1
                dp[0][i] = (x, y)
                if x == y and x > 0:
                    result += 1
        #  Populate first column
        for i in range(1, m):
            x, y = dp[i - 1][0]
            if grid[i][0] == "X":
                x += 1
            elif grid[i][0] == "Y":
                y += 1
            dp[i][0] = (x, y)
            if x == y and x > 0:
                result += 1
        for i in range(1, m):
            for j in range(1, n):
                x, y = dp[i - 1][j]
                x += dp[i][j-1][0]
                y += dp[i][j-1][1]
                x -= dp[i -1][j - 1][0]
                y -= dp[i -1][j - 1][1]
                if grid[i][j] == "X":
                    x += 1
                elif grid[i][j] == "Y":
                    y += 1
                dp[i][j] = (x, y)
                if x == y and x > 0:
                    result += 1
        return result
                    
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