---
tag: EX382NQTGP
kind: concept
name: Enumerative combinatorics
msc: [05A15, 05A16, 05A99]
specializes: [0JKW7W7XFR]
sources:
  - url: https://en.wikipedia.org/wiki/Enumerative_combinatorics
same_as:
  wikidata: Q336787
  wikipedia: Enumerative combinatorics
---

## Summary

Enumerative combinatorics is the part of [[combinatorics|0JKW7W7XFR]] that counts finite structures, see the [[definition|NWGARTY61Y]].
It asks for the number of objects of size $n$ in a family, as a function of $n$.
Its answers take the form of closed formulas, [[recurrence relations|0FYX44GP92]], [[generating functions|0PHAC833JF]] or asymptotic estimates.
Its tools range from bijections and the inclusion-exclusion principle to complex analysis in [[analytic combinatorics|YJK71RKVEB]].

## Explanations

### L1 human

Enumerative combinatorics answers questions that begin with "how many".
How many ways are there to climb $n$ stairs taking one or two steps at a time?
The answer is a Fibonacci number, and a recurrence finds it: the last step is a single or a double step.
The aim is a formula or a fast method that works for every $n$ at once, not a list of cases.