Möbius inversion theorem

The M\"obius inversion theorem, proven by August Ferdinand M\"obius in 1832, is a mathematical principle in combinatorics. It states that if a function f (defined on positive integers) is a sum of values of another function g evaluated at the divisors of its input, then g can also be expressed as a sum involving f evaluated at the divisors of its input.1

A powerful generalization of this theorem was developed by Gian-Carlo Rota in 1964, unifying enumeration principles. This generalization applies to functions defined on subsets of a finite set, allowing one function to be expressed in terms of another.1

It is important to distinguish this theorem from the M\"obius strip, a geometric surface with one side and one boundary, discovered independently by M\"obius and Johann Benedict Listing in 1858.2