Count Submatrices With Equal Frequency of X and Y

For the usual version, each submatrix is anchored at the top-left corner, so a two-dimensional prefix scan is enough. For every cell, derive the number of X and Y characters in rectangle (0,0) through (r,c) from prior row/column prefixes, then count it when X equals Y and the count of X is positive. Equivalently track a balance X minus Y, but keep a separate presence count because a rectangle containing neither character has balance zero yet is not valid. Be explicit about whether dots/other characters contribute zero and whether only anchored rectangles are requested; arbitrary submatrices require a more involved row-pair reduction. The anchored DP is O(RC) time and O(RC) space, or O(C) space with rolling rows.