Exponential generating function

A formal power series of the form ∑ₙ aₙxⁿ/n!, encoding a sequence with each coefficient weighted by the reciprocal factorial.

Exponential formula: A theorem relating labeled structures assembled from connected components to their exponential generating functions. It explains why sets of labeled components correspond to exponentiating a component series.
Labeled tree: A tree whose vertices carry distinct labels. Exponential generating functions encode counts of labeled trees by vertex number.
Factorial: For a nonnegative integer n, the product of the positive integers up to n, with 0! equal to 1. The denominator n! is the defining weight on the nth coefficient.
Arthur Cayley: A nineteenth-century British mathematician known for foundational work in algebra and graph theory. Cayley counted labeled trees, a classic problem later expressed through exponential generating functions.
Dirichlet generating function: A series of the form ∑ₙ aₙn⁻ˢ that encodes a sequence using powers of integer indices. It weights terms by n⁻ˢ rather than powers of a formal variable divided by factorials.
Labeled combinatorial class: A collection of structures built on finite label sets, where relabeling preserves the structures. Exponential generating functions naturally count structures whose distinct labels matter.
Derangement: A permutation with no fixed points. Its exponential generating function follows from excluding fixed points in labeled permutations.
Sequence: An ordered list of values indexed by integers, often nonnegative integers. The series packages a sequence into a single algebraic object.
James Joseph Sylvester: A British mathematician whose work helped establish invariant theory and combinatorics. Sylvester developed enumerative methods in which generating functions became a key tool.
Ordinary generating function: A power series ∑ₙ aₙxⁿ that encodes a sequence directly as its coefficients. It is the closest competing convention, suited especially to unlabeled constructions.

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