Math 2.0

Combinatorics Definition

Enumerative combinatorics

Builds on CombinatoricsFinite setGenerating functionRecurrence relation

Definition

Let (Sn)n≥0(S_n)_{n \ge 0} be a family of finite sets given by a combinatorial rule, such as the subsets of {1,…,n}\set{1, \ldots, n} or the trees with nn vertices. Enumerative combinatorics is the branch of combinatorics that determines the counting function n↦∣Sn∣n \mapsto \abs{S_n}. An answer is one of the following:

  1. an explicit formula for ∣Sn∣\abs{S_n};
  2. a recurrence relation that computes ∣Sn∣\abs{S_n} from earlier values;
  3. a closed form for a generating function of the sequence ∣Sn∣\abs{S_n};
  4. an asymptotic estimate of ∣Sn∣\abs{S_n} as n→∞n \to \infty.

Remarks

Example: let SnS_n be the set of subsets of {1,…,n}\set{1, \ldots, n} with no two consecutive elements. A set in SnS_n either avoids nn, or contains nn and avoids n−1n - 1. This gives the recurrence ∣Sn∣=∣Sn−1∣+∣Sn−2∣\abs{S_n} = \abs{S_{n-1}} + \abs{S_{n-2}}, with ∣S0∣=1\abs{S_0} = 1 and ∣S1∣=2\abs{S_1} = 2, so ∣Sn∣\abs{S_n} is the Fibonacci number Fn+2F_{n+2}.

Non-example: the question whether a finite projective plane of order 1010 exists is a question of existence, not of counting. It belongs to the theory of combinatorial designs.

The parameter nn may be replaced by several parameters, as in the binomial coefficient (nk)\binom{n}{k}.

Reading path

To understand this page from the ground up, read these 23 pages in order. The foundations come first.

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Combinatorics, concept, no evidence levelCombinatoricsconceptElement, definition, no evidence levelElementdefinitionGenerating function, concept, no evidence levelGenerating functionconceptProposition, concept, no evidence levelPropositionconceptRecurrence relation, concept, no evidence levelRecurrence relationconceptSet, definition, no evidence levelSetdefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionFunction, definition, no evidence levelFunctiondefinitionCodomain, definition, no evidence levelCodomaindefinitionInjective function, definition, no evidence levelInjective functiondefinitionSurjective function, definition, no evidence levelSurjective functiondefinitionBijective function, definition, no evidence levelBijective functiondefinitionEquinumerous sets, definition, no evidence levelEquinumerous setsdefinitionFinite set, definition, no evidence levelFinite setdefinitionEnumerative combinatorics, definition, no evidence levelEnumerative combinato…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Combinatorics concept
  2. Element definition
  3. Generating function concept
  4. Proposition concept
  5. Recurrence relation concept
  6. Set definition
  7. Implication definition
  8. Predicate definition
  9. Ordered pair definition
  10. Existential quantifier definition
  11. Universal quantifier definition
  12. Cartesian product definition
  13. Subset definition
  14. Uniqueness quantifier definition
  15. Binary relation definition
  16. Function definition
  17. Codomain definition
  18. Injective function definition
  19. Surjective function definition
  20. Bijective function definition
  21. Equinumerous sets definition
  22. Finite set definition
  23. Enumerative combinatorics definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Enumerative_combinatorics
About this page
TagNWGARTY61Y
URIm2://NWGARTY61Y
Kinddefinition
WritingW0 stub
MSC05A15, 05A16, 05A99
DefinesEnumerative combinatorics
UsesCombinatorics, Finite set, Generating function, Recurrence relation
Changed2 November 2026

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