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different partitions of staircase of height n

Source: Russian Regional Olympiad 2010 10.8

August 24, 2024
combinatoricscombinatorial geometry

Problem Statement

Let's call it a staircase of height nn, a figure consisting from all square cells n×nn\times n lying no higher diagonals (the figure shows a staircase of height 44 ). In how many different ways can a staircase of height nn can be divided into several rectangles whose sides go along the grid lines, but the areas are different in pairs? https://cdn.artofproblemsolving.com/attachments/f/0/f66d7e9ada0978e8403fbbd8989dc1b201f2cd.png