Bell numbers
E586577
Bell numbers are a sequence in combinatorics that count the number of ways to partition a finite set into nonempty, unlabeled subsets.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Bell numbers canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T6327847 — 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: Bell numbers Context triple: [enumerative combinatorics, usesConcept, Bell 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.
Bernoulli numbers
Bernoulli numbers are a sequence of rational numbers that play a central role in number theory and analysis, especially in formulas for sums of powers of integers and in the study of special functions like the Riemann zeta function.
-
D.
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.
-
E.
Pochhammer symbol
The Pochhammer symbol is a mathematical notation representing rising factorials, widely used in series expansions, special functions, and hypergeometric functions.
- 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: Bell numbers Target entity description: Bell numbers are a sequence in combinatorics that count the number of ways to partition a finite set into nonempty, unlabeled subsets.
-
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.
Bernoulli numbers
Bernoulli numbers are a sequence of rational numbers that play a central role in number theory and analysis, especially in formulas for sums of powers of integers and in the study of special functions like the Riemann zeta function.
-
D.
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.
-
E.
Pochhammer symbol
The Pochhammer symbol is a mathematical notation representing rising factorials, widely used in series expansions, special functions, and hypergeometric functions.
- F. None of above. chosen
Statements (50)
| Predicate | Object |
|---|---|
| instanceOf |
combinatorial sequence
ⓘ
integer sequence ⓘ |
| alternativeNotation | Bell(n) ⓘ |
| application |
counting possible clusterings of data points
ⓘ
counting possible set partitions in combinatorial optimization ⓘ enumeration in partition-based probability models ⓘ |
| asymptoticGrowth | B_n \sim \frac{1}{\sqrt{n}} \left(\frac{n}{W(n)}\right)^{n+1/2} e^{\frac{n}{W(n)}-n-1} ⓘ |
| B0 | 1 ⓘ |
| B1 | 1 ⓘ |
| B10 | 115975 ⓘ |
| B2 | 2 ⓘ |
| B3 | 5 ⓘ |
| B4 | 15 ⓘ |
| B5 | 52 ⓘ |
| B6 | 203 ⓘ |
| B7 | 877 ⓘ |
| B8 | 4140 ⓘ |
| B9 | 21147 ⓘ |
| combinatorialInterpretation |
number of equivalence relations on an n-element set
ⓘ
number of ways to partition an n-element set into any number of nonempty blocks ⓘ |
| definition | number of set partitions of an n-element set into nonempty unlabeled subsets ⓘ |
| DobinskiFormula | B_n = \frac{1}{e} \sum_{k=0}^{\infty} \frac{k^n}{k!} ⓘ |
| eighthTerm | 877 ⓘ |
| exponentialGeneratingFunction | \sum_{n=0}^{\infty} B_n \frac{x^n}{n!} = e^{e^x - 1} ⓘ |
| field | combinatorics ⓘ |
| fifthTerm | 15 ⓘ |
| firstTerm | 1 ⓘ |
| fourthTerm | 5 ⓘ |
| growthType | superexponential ⓘ |
| matrixRepresentation | can be computed via the Bell triangle (Aitken array) ⓘ |
| namedAfter | Eric Temple Bell ⓘ |
| ninthTerm | 4140 ⓘ |
| OEISID | A000110 ⓘ |
| parityProperty | B_n is odd iff n has no 2s in its binary expansion ⓘ |
| property |
B_n is integer for all nonnegative integers n
ⓘ
B_n is strictly increasing for n \ge 1 ⓘ |
| recurrenceRelation |
B_n = \sum_{k=0}^{n} S(n,k)
ⓘ
B_{n+1} = \sum_{k=0}^{n} \binom{n}{k} B_k ⓘ B_{n} = \sum_{k=1}^{n} \binom{n-1}{k-1} B_{n-k} ⓘ |
| relatedTo |
Bell polynomials
ⓘ
Stirling numbers of the second kind ⓘ Touchard polynomials ⓘ
linked to:
Sheffer sequences
equivalence relations ⓘ set partitions ⓘ |
| secondTerm | 1 ⓘ |
| seventhTerm | 203 ⓘ |
| sixthTerm | 52 ⓘ |
| symbol | B_n ⓘ |
| tenthTerm | 21147 ⓘ |
| thirdTerm | 2 ⓘ |
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: Bell numbers Description of subject: Bell numbers are a sequence in combinatorics that count the number of ways to partition a finite set into nonempty, unlabeled subsets.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.