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11: 26.7 Set Partitions: Bell Numbers

§26.7 Set Partitions: Bell Numbers

►

§26.7(i) Definitions

►

B\left(n\right)

is the number of partitions of

\{1,2,\ldots,n\}

. … ►

§26.7(ii) Generating Function

… ►

§26.7(iii) Recurrence Relation

…

12: 24.19 Methods of Computation

… ►

§24.19(i) Bernoulli and Euler Numbers and Polynomials

… ►A similar method can be used for the Euler numbers based on (4.19.5). … ►

§24.19(ii) Values of $B_{n}$ Modulo $p$

►For number-theoretic applications it is important to compute

B_{2n}\pmod{p}

for

2n\leq p-3

; in particular to find the irregular pairs

(2n,p)

for which

B_{2n}\equiv 0\pmod{p}

. … ►

•

Buhler et al. (1992) uses the expansion

24.19.3

\frac{t^{2}}{\cosh t-1}=-2\sum_{n=0}^{\infty}(2n-1)B_{2n}\frac{t^{2n}}{(2n)!},

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $!$ : factorial (as in $n!$ ), $\cosh\NVar{z}$ : hyperbolic cosine function, $n$ : integer and $t$ : real or complex
Referenced by:: §24.19(ii)
Permalink:: http://dlmf.nist.gov/24.19.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §24.19(ii), §24.19 and Ch.24

and computes inverses modulo $p$ of the left-hand side. Multisectioning techniques are applied in implementations. See also Crandall (1996, pp. 116–120).

…

13: 27.2 Functions

… ►where

p_{1},p_{2},\dots,p_{\nu\left(n\right)}

are the distinct prime factors of

n

, each exponent

a_{r}

is positive, and

\nu\left(n\right)

is the number of distinct primes dividing

n

. … ►(See Gauss (1863, Band II, pp. 437–477) and Legendre (1808, p. 394).) … ►The

\phi\left(n\right)

numbers

a,a^{2},\dots,a^{\phi\left(n\right)}

are relatively prime to

n

and distinct (mod

n

). … … ►

§27.2(ii) Tables

…

14: 24.10 Arithmetic Properties

§24.10 Arithmetic Properties

… ►The denominator of

B_{2n}

is the product of all these primes

p

. … ►

§24.10(ii) Kummer Congruences

… ►

§24.10(iii) Voronoi’s Congruence

… ►

§24.10(iv) Factors

…

15: 26.8 Set Partitions: Stirling Numbers

§26.8 Set Partitions: Stirling Numbers

►

§26.8(i) Definitions

►

s\left(n,k\right)

denotes the Stirling number of the first kind:

(-1)^{n-k}

times the number of permutations of

\{1,2,\ldots,n\}

with exactly

k

cycles. … ►

S\left(n,k\right)

denotes the Stirling number of the second kind: the number of partitions of

\{1,2,\ldots,n\}

into exactly

k

nonempty subsets. … ►

§26.8(vi) Relations to Bernoulli Numbers

…

16: 24.14 Sums

§24.14 Sums

►

§24.14(i) Quadratic Recurrence Relations

… ►

§24.14(ii) Higher-Order Recurrence Relations

►In the following two identities, valid for

n\geq 2

, the sums are taken over all nonnegative integers

j,k,\ell

with

j+k+\ell=n

. … ►For other sums involving Bernoulli and Euler numbers and polynomials see Hansen (1975, pp. 331–347) and Prudnikov et al. (1990, pp. 383–386).

17: 26.21 Tables

§26.21 Tables

►Abramowitz and Stegun (1964, Chapter 24) tabulates binomial coefficients

\genfrac{(}{)}{0.0pt}{}{m}{n}

for

m

up to 50 and

n

up to 25; extends Table 26.4.1 to

n=10

; tabulates Stirling numbers of the first and second kinds,

s\left(n,k\right)

and

S\left(n,k\right)

, for

n

up to 25 and

k

up to

n

; tabulates partitions

p\left(n\right)

and partitions into distinct parts

p\left(\mathcal{D},n\right)

for

n

up to 500. ►Andrews (1976) contains tables of the number of unrestricted partitions, partitions into odd parts, partitions into parts

\not\equiv\pm 2\pmod{5}

, partitions into parts

\not\equiv\pm 1\pmod{5}

, and unrestricted plane partitions up to 100. It also contains a table of Gaussian polynomials up to

\genfrac{[}{]}{0.0pt}{}{12}{6}_{q}

. ►Goldberg et al. (1976) contains tables of binomial coefficients to

n=100

and Stirling numbers to

n=40

18: 26.13 Permutations: Cycle Notation

… ►The number of elements of

\mathfrak{S}_{n}

with cycle type

{\left(a_{1},a_{2},\ldots,a_{n}\right)}

is given by (26.4.7). ►The Stirling cycle numbers of the first kind, denoted by

\genfrac{[}{]}{0.0pt}{}{n}{k}

, count the number of permutations of

\{1,2,\ldots,n\}

with exactly

k

cycles. They are related to Stirling numbers of the first kind by …See §26.8 for generating functions, recurrence relations, identities, and asymptotic approximations. … ►The derangement number,

d(n)

, is the number of elements of

\mathfrak{S}_{n}

with no fixed points: …

19: 26.1 Special Notation

… ► ►►►►►

$\genfrac{(}{)}{0.0pt}{}{m}{n}$	binomial coefficient.
…
$\genfrac{<}{>}{0.0pt}{}{m}{n}$	Eulerian number.
…
$B\left(n\right)$	Bell number.
$C\left(n\right)$	Catalan number.
…

… ►Other notations for

s\left(n,k\right)

, the Stirling numbers of the first kind, include

S_{n}^{(k)}

(Abramowitz and Stegun (1964, Chapter 24), Fort (1948)),

S_{n}^{k}

(Jordan (1939), Moser and Wyman (1958a)),

\genfrac{(}{)}{0.0pt}{}{n-1}{k-1}B_{n-k}^{(n)}

(Milne-Thomson (1933)),

(-1)^{n-k}S_{1}(n-1,n-k)

(Carlitz (1960), Gould (1960)),

(-1)^{n-k}\left[n\atop k\right]

(Knuth (1992), Graham et al. (1994), Rosen et al. (2000)). ►Other notations for

S\left(n,k\right)

, the Stirling numbers of the second kind, include

\mathscr{S}^{(k)}_{n}

(Fort (1948)),

\mathfrak{S}_{n}^{k}

(Jordan (1939)),

\sigma_{n}^{k}

(Moser and Wyman (1958b)),

\genfrac{(}{)}{0.0pt}{}{n}{k}B_{n-k}^{(-k)}

(Milne-Thomson (1933)),

S_{2}(k,n-k)

(Carlitz (1960), Gould (1960)),

\left\{n\atop k\right\}

(Knuth (1992), Graham et al. (1994), Rosen et al. (2000)), and also an unconventional symbol in Abramowitz and Stegun (1964, Chapter 24).

20: 24.5 Recurrence Relations

§24.5 Recurrence Relations

… ►

24.5.5

\sum_{k=0}^{n}{n\choose k}2^{k}E_{n-k}+E_{n}=2.

ⓘ

Symbols:: $E_{\NVar{n}}$ : Euler numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $k$ : integer and $n$ : integer
Permalink:: http://dlmf.nist.gov/24.5.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §24.5(i), §24.5 and Ch.24

►

§24.5(ii) Other Identities

… ►

24.5.8

\sum_{k=0}^{n}\frac{2^{2k}B_{2k}}{(2k)!(2n+1-2k)!}=\frac{1}{(2n)!},

n=1,2,\dots

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $!$ : factorial (as in $n!$ ), $k$ : integer and $n$ : integer
Referenced by:: §24.5(ii)
Permalink:: http://dlmf.nist.gov/24.5.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §24.5(ii), §24.5 and Ch.24

►

§24.5(iii) Inversion Formulas

…

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11—20 of 837 matching pages

11: 26.7 Set Partitions: Bell Numbers

§26.7 Set Partitions: Bell Numbers

§26.7(i) Definitions

§26.7(ii) Generating Function

§26.7(iii) Recurrence Relation

12: 24.19 Methods of Computation

§24.19(i) Bernoulli and Euler Numbers and Polynomials

§24.19(ii) Values of B n Modulo p

13: 27.2 Functions

§27.2(ii) Tables

14: 24.10 Arithmetic Properties

§24.10 Arithmetic Properties

§24.10(ii) Kummer Congruences

§24.10(iii) Voronoi’s Congruence

§24.10(iv) Factors

15: 26.8 Set Partitions: Stirling Numbers

§26.8 Set Partitions: Stirling Numbers

§26.8(i) Definitions

§26.8(vi) Relations to Bernoulli Numbers

16: 24.14 Sums

§24.14 Sums

§24.14(i) Quadratic Recurrence Relations

§24.14(ii) Higher-Order Recurrence Relations

17: 26.21 Tables

§26.21 Tables

18: 26.13 Permutations: Cycle Notation

19: 26.1 Special Notation

20: 24.5 Recurrence Relations

§24.5 Recurrence Relations

§24.5(ii) Other Identities

§24.5(iii) Inversion Formulas

§24.19(ii) Values of $B_{n}$ Modulo $p$