Question 123
In a square grid $n \times n$ $n$-staircase (4-staircase is shown in the figure). find the number of ways to partition an $n$-staircase (D. khramtsov) 
Let's call the height stair $n$, consisting of all cells of the square $n×n$, which lie no higher than the diagonal (Figure shows staircase height 4). How many different ways can you break the $n$ height ladder into several rectangles, the sides of which go along the grid lines, and the areas are different in pairs? 
Let's call the height stair $n$, consisting of all cells of the square $n×n$, which lie no higher than the diagonal (Figure shows staircase height 4). How many different ways can you break the $n$ height ladder into several rectangles, the sides of which go along the grid lines, and the areas are different in pairs?