BellNumber

The nth Bell number, counting the ways to partition a set of n elements.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaBell number↗MathWorldBellNumber↗WikidataQ816063↗DLMF26.7.i↗FungrimBellNumber✓↗Fungrim entry60dc3e✓↗f4e249✓↗OEISA000110↗Wolfram LanguageBellB!↗

BellNumber(n)the nth Bell number $B_n$, the number of partitions of an n-element set.

SageMathbell_number(n)✓↗SymPybell(n)↗Oscarbell(n)Juliabell(ZZ(n))

BellNumber(n, x)the Bell (Touchard) polynomial $B_n(x) = \sum_k S(n,k)\,x^k$, exact for a nonnegative integer n.

Typed signatures

Type	Package	Applies
(integer) -> integer	compute-engine	always
(integer, any?) -> any	number-theoryreplaces compute-engine	always

Domain: Combinatorics

Details

• Recurrence $B_{n+1} = \sum_{k=0}^{n} \binom{n}{k}\, B_k$, built from Binomial and the lower Bell numbers.
• Generating function $\sum_{n} B_n \frac{t^n}{n!} = e^{e^t - 1}$.
• Equals the sum of Stirling numbers of the second kind over all block counts, $B_n = \sum_{k=0}^{n} S(n, k)$.
• Also arise as the nth moment of a Poisson distribution with mean 1.
• compute-engine requires a nonnegative integer argument; a negative n is left unevaluated.
• The two-argument form $B_n(x)$ is the Bell (Touchard) polynomial; $B_n(1) = B_n$.

Examples

1 more kept as data, checked against the oracles

Basic #

live ↗

In$\mathrm{BellNumber}(0)$Out$1$

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In$\mathrm{BellNumber}(1)$Out$1$

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In$\mathrm{BellNumber}(3)$Out$5$

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In$\mathrm{BellNumber}(5)$Out$52$

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In$\mathrm{BellNumber}(10)$Out$115\,975$

Scope #

Threads element-wise over a list, as Wolfram's Listable heads do

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In$\mathrm{BellNumber}(\bigl\lbrack1, 2, 3, 4\bigr\rbrack)$Out$\bigl\lbrack1, 2, 5, 15\bigr\rbrack$

Exact for large n

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In$\mathrm{BellNumber}(30)$Out$846\,749\,014\,511\,809\,332\,450\,147$

The two-argument form gives the Bell (Touchard) polynomial $\sum_k S(n, k)\,x^k$

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In$\mathrm{BellNumber}(5, x)$Out$x^5+10x^4+25x^3+15x^2+x$

The Bell polynomial at a real point, $B_{20}(0.5)$

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In$\mathrm{BellNumber}(20, 0.5)$Out$270\,190\,131\,940.151\,92$

Threads over a list of indices

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In$\mathrm{BellNumber}(\bigl\lbrack1, 2, 3, 4, 5, 6\bigr\rbrack)$Out$\bigl\lbrack1, 2, 5, 15, 52, 203\bigr\rbrack$

Applications #

The number of ways to partition a 4-element set into non-empty, unlabeled subsets

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In$\mathrm{BellNumber}(4)$Out$15$

Properties #

Recurrence: $B_{n+1} = \sum_{k=0}^n \binom{n}{k} B_k$, here at $n = 4$

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In$\mathrm{BellNumber}(5)=\binom{4}{0}\mathrm{BellNumber}(0)+\binom{4}{1}\mathrm{BellNumber}(1)+\binom{4}{2}\mathrm{BellNumber}(2)+\binom{4}{3}\mathrm{BellNumber}(3)+\binom{4}{4}\mathrm{BellNumber}(4)$Out$\mathrm{True}$

The Bell polynomial at 1 is the Bell number: $B_n(1) = B_n$

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In$\mathrm{BellNumber}(5, 1)$Out$52$

Dobinski's formula $B_n = \frac{1}{e}\sum_{k \ge 0} \frac{k^n}{k!}$

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In$\frac{1}{e}(\sum_{k=0}^{∞}\frac{k^5}{k!})$Out$52$

Touchard's congruence $B_{p+n} \equiv B_n + B_{n+1} \pmod p$, here $p = 7$, $n = 2$

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In$\mathrm{BellNumber}(9)\bmod7=(\mathrm{BellNumber}(2)+\mathrm{BellNumber}(3))\bmod7$Out$\mathrm{True}$

Exponential generating function is $e^{e^x-1}$

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In$\mathrm{ExponentialGeneratingFunction}(\mathrm{BellNumber}(n), n, x)$Out$e^{e^{x}-1}$

Two elements split into set partitions in 2 ways

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In$\mathrm{BellNumber}(2)$Out$2$

Known value · Wolfram MathematicalFunctionData, BellB

203 set partitions of a 6-element set

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In$\mathrm{BellNumber}(6)$Out$203$

Known value · Wolfram MathematicalFunctionData, BellB

877 set partitions of a 7-element set

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In$\mathrm{BellNumber}(7)$Out$877$

Known value · Wolfram MathematicalFunctionData, BellB

Growth is fast: 4140 partitions at $n=8$

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In$\mathrm{BellNumber}(8)$Out$4\,140$

Known value · Wolfram MathematicalFunctionData, BellB

21147 set partitions of 9 elements

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In$\mathrm{BellNumber}(9)$Out$21\,147$

Known value · Wolfram MathematicalFunctionData, BellB

The Bell polynomial $B_1(x)=x$ vanishes at 0

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In$\mathrm{BellNumber}(1, 0)$Out$0$

Known value · Wolfram MathematicalFunctionData, BellB:Polynomial

The Bell polynomial $B_2(x)=x^2+x$ vanishes at 0

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In$\mathrm{BellNumber}(2, 0)$Out$0$

Known value · Wolfram MathematicalFunctionData, BellB:Polynomial

Possible issues #

A negative argument is left unevaluated

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In$\mathrm{BellNumber}(-1)$Out$\mathrm{BellNumber}(-1)$

Neat examples #

The first ten Bell numbers

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In$\mathrm{BellNumber}(1..10)$Out$\bigl\lbrack1, 2, 5, 15, 52, 203, 877, 4\,140, 21\,147, 115\,975\bigr\rbrack$

About

Examples	26
Checked against	Wolfram Language 25 agree1 differSageMath 17 agree2 errorSymPy 17 agree2 errormathlib4 14 agree5 errorOscar 8 agree4 errorJulia 16 agree2 error
Crosswalk	WikipediaMathWorldWikidataDLMFFungrimFungrim entry ×2OEISWolfram LanguageSageMathSymPyOscarJulia

Implementation

mappedwolframpackages/oracle/src/mappings.ts

Wolfram spells it BellB. Julia's bare bell(::Int) tries to fit an arbitrary-precision result into an Int64 past n≈25; ZZ keeps it exact.

mappedsympypackages/oracle/src/mappings.ts

Wolfram spells it BellB. Julia's bare bell(::Int) tries to fit an arbitrary-precision result into an Int64 past n≈25; ZZ keeps it exact.

mappedsagepackages/oracle/src/mappings.ts

Wolfram spells it BellB. Julia's bare bell(::Int) tries to fit an arbitrary-precision result into an Int64 past n≈25; ZZ keeps it exact.

mappedoscarpackages/oracle/src/mappings.ts

Wolfram spells it BellB. Julia's bare bell(::Int) tries to fit an arbitrary-precision result into an Int64 past n≈25; ZZ keeps it exact.

mappedjuliapackages/oracle/src/mappings.ts

Wolfram spells it BellB. Julia's bare bell(::Int) tries to fit an arbitrary-precision result into an Int64 past n≈25; ZZ keeps it exact.

mappedmathlib4packages/oracle/src/mappings.ts

Nat.bell counts the partitions of an n-element set, B(0) = 1; natural n only.

See also: StirlingS1