Enumeration of tilings is the mathematical study concerning the total number of ways to cover a certain region by similar pieces so that there are no gaps or overlaps. The study dates back to the early 1900s when MacMahon's proved his classical theorem about boxed plane partitions, which are in bijection with lozenge tilings of a hexagon. Enumeration of tilings has become a vibrant subfield in algebraic and enumeration combinatorics with applications and connections to various areas, including symmetric functions, statistical mechanics, cluster algebra, and probability. We also talk about a powerful method of enumerating tilings, namely Kuo's graphical condensation.
