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.
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 NERFINISHED ⓘ |
| 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 NERFINISHED ⓘ equivalence relations ⓘ set partitions ⓘ |
| secondTerm | 1 ⓘ |
| seventhTerm | 203 ⓘ |
| sixthTerm | 52 ⓘ |
| symbol | B_n ⓘ |
| tenthTerm | 21147 ⓘ |
| thirdTerm | 2 ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.