Exponential generating function

An exponential generating function (EGF) is a mathematical tool that encodes an infinite sequence of numbers — like counts of objects or arrangements — into a single formula. Unlike standard generating functions, each term in an EGF is divided by the factorial of its position, which is especially useful in combinatorics when dealing with labeled structures. This approach simplifies calculations involving permutations, arrangements, or labeled graphs, making it easier to analyze and find patterns within complex sequences. Essentially, the EGF provides a compact, powerful way to study sequences and their properties through calculus and algebra.