Catalan numbers
E586576
Catalan numbers are a sequence of natural numbers that count a wide variety of combinatorial structures, such as correctly matched parentheses, binary tree shapes, and lattice path configurations.
All labels observed (2)
| Label | Occurrences |
|---|---|
| A000108 | 1 |
| Catalan numbers canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6327844 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Catalan numbers Context triple: [enumerative combinatorics, usesConcept, Catalan numbers]
-
A.
The Twelvefold Way
The Twelvefold Way is a framework in combinatorics that systematically classifies twelve fundamental ways of counting functions between finite sets under various labeling and structural constraints.
-
B.
Delannoy
Delannoy is a French surname, often associated with historical figures and families of French or Flemish origin.
-
C.
Pascal's triangle
Pascal's triangle is a triangular array of numbers in which each entry is the sum of the two directly above it, widely used in combinatorics, algebra, and probability.
-
D.
enumerative combinatorics
Enumerative combinatorics is a branch of mathematics focused on counting and characterizing discrete structures, often using generating functions, bijections, and algebraic techniques.
-
E.
Pascal's identity
Pascal's identity is a fundamental combinatorial formula that relates adjacent binomial coefficients and underlies many proofs and properties of binomial expansions.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Catalan numbers Target entity description: Catalan numbers are a sequence of natural numbers that count a wide variety of combinatorial structures, such as correctly matched parentheses, binary tree shapes, and lattice path configurations.
-
A.
The Twelvefold Way
The Twelvefold Way is a framework in combinatorics that systematically classifies twelve fundamental ways of counting functions between finite sets under various labeling and structural constraints.
-
B.
Delannoy
Delannoy is a French surname, often associated with historical figures and families of French or Flemish origin.
-
C.
Pascal's triangle
Pascal's triangle is a triangular array of numbers in which each entry is the sum of the two directly above it, widely used in combinatorics, algebra, and probability.
-
D.
enumerative combinatorics
Enumerative combinatorics is a branch of mathematics focused on counting and characterizing discrete structures, often using generating functions, bijections, and algebraic techniques.
-
E.
Pascal's identity
Pascal's identity is a fundamental combinatorial formula that relates adjacent binomial coefficients and underlies many proofs and properties of binomial expansions.
- F. None of above. chosen
Statements (50)
| Predicate | Object |
|---|---|
| instanceOf |
combinatorial sequence
ⓘ
integer sequence ⓘ |
| asymptoticGrowth | C_n ~ 4^n / (n^{3/2} * sqrt(pi)) ⓘ |
| counts |
number of Dyck paths of semilength n
ⓘ
number of correct bracket sequences of length 2n ⓘ number of full binary tree shapes with n+1 leaves ⓘ number of monotonic lattice paths along grid edges from (0,0) to (n,n) that do not cross above the diagonal y=x ⓘ number of non-crossing partitions of an n-element set ⓘ number of rooted ordered binary trees with n internal nodes ⓘ number of stack-sortable permutations of length n ⓘ number of standard Young tableaux of shape (n,n) ⓘ number of ways to insert parentheses in a product of n+1 factors ⓘ number of ways to triangulate a convex (n+2)-gon ⓘ |
| fieldOfStudy |
combinatorics
ⓘ
enumerative combinatorics ⓘ |
| generalTermFormula |
C_n = (1/(n+1)) * binomial(2n, n)
ⓘ
C_n = (2n)! / ((n+1)! n!) ⓘ C_n = binomial(2n, n) - binomial(2n, n+1) ⓘ |
| generatingFunction | C(x) = (1 - sqrt(1-4x)) / (2x) ⓘ |
| growthType | exponential ⓘ |
| hasEighthTerm | 429 ⓘ |
| hasFifthTerm | 14 ⓘ |
| hasFirstTerm | 1 ⓘ |
| hasFourthTerm | 5 ⓘ |
| hasNinthTerm | 1430 ⓘ |
| hasSecondTerm | 1 ⓘ |
| hasSeventhTerm | 132 ⓘ |
| hasSixthTerm | 42 ⓘ |
| hasTenthTerm | 4862 ⓘ |
| hasThirdTerm | 2 ⓘ |
| indexDomain | n ≥ 0 ⓘ |
| monotonicity | C_n is strictly increasing for n ≥ 1 ⓘ |
| namedAfter | Eugène Charles Catalan ⓘ |
| nonNegativity | C_n ≥ 0 for all n ≥ 0 ⓘ |
| OEISID |
A000108
ⓘ
linked to:
Catalan numbers
|
| recurrenceRelation |
C_0 = 1
ⓘ
C_{n+1} = (4n+2)/(n+2) * C_n ⓘ C_{n+1} = sum_{i=0}^{n} C_i C_{n-i} ⓘ |
| relatedConcept |
Dyck paths
ⓘ
ballot problem ⓘ binary trees ⓘ non-crossing partitions ⓘ |
| relatedSequence |
Motzkin numbers
ⓘ
Schröder numbers ⓘ |
| valueAtFive | C_5 = 42 ⓘ |
| valueAtFour | C_4 = 14 ⓘ |
| valueAtOne | C_1 = 1 ⓘ |
| valueAtThree | C_3 = 5 ⓘ |
| valueAtTwo | C_2 = 2 ⓘ |
| valueAtZero | C_0 = 1 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Catalan numbers Description of subject: Catalan numbers are a sequence of natural numbers that count a wide variety of combinatorial structures, such as correctly matched parentheses, binary tree shapes, and lattice path configurations.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Catalan numbers