Binary Subarrays With Sum / visualized

You cannot slide a window for exactly goal. The predicate is not monotone: extending the window to the right can carry you from sum == goal to sum > goal, and there is no shrink rule that lands you back — shrinking from the left can overshoot straight past the target. But sum ≤ s is monotone: once a window is too heavy, dropping elements only ever helps, so atMost(s) is windowable. That gives the whole sub-variant its identity: exactly(goal) = atMost(goal) − atMost(goal−1). Run the same O(n) window twice and subtract. Inside each pass, a valid window [l..r] banks r − l + 1 subarrays at once — one per suffix ending at r. Watch the edge: when goal = 0 the second pass is atMost(−1), which must be defined as 0, not run.