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11: 26.5 Lattice Paths: Catalan Numbers

§26.5 Lattice Paths: Catalan Numbers

►

§26.5(i) Definitions

►

C\left(n\right)

is the Catalan number. … ►

§26.5(ii) Generating Function

… ►

§26.5(iii) Recurrence Relations

…

12: 26.14 Permutations: Order Notation

… ►As an example,

35247816

is an element of

\mathfrak{S}_{8}.

The inversion number is the number of pairs of elements for which the larger element precedes the smaller: … ► ►The Eulerian number, denoted

\genfrac{<}{>}{0.0pt}{}{n}{k}

, is the number of permutations in

\mathfrak{S}_{n}

with exactly

k

descents. …The Eulerian number

\genfrac{<}{>}{0.0pt}{}{n}{k}

is equal to the number of permutations in

\mathfrak{S}_{n}

with exactly

k

excedances. … ►

§26.14(iii) Identities

…

13: 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

… ►

§26.7(iv) Asymptotic Approximation

…

14: 26.8 Set Partitions: Stirling Numbers

§26.8 Set Partitions: Stirling Numbers

►

§26.8(i) Definitions

… ► … ►

§26.8(v) Identities

… ►

§26.8(vi) Relations to Bernoulli Numbers

…

15: 24.19 Methods of Computation

… ►

§24.19(i) Bernoulli and Euler Numbers and Polynomials

►Equations (24.5.3) and (24.5.4) enable

B_{n}

and

E_{n}

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

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

… ►We list here three methods, arranged in increasing order of efficiency. …

16: 27.17 Other Applications

§27.17 Other Applications

►Reed et al. (1990, pp. 458–470) describes a number-theoretic approach to Fourier analysis (called the arithmetic Fourier transform) that uses the Möbius inversion (27.5.7) to increase efficiency in computing coefficients of Fourier series. ►Congruences are used in constructing perpetual calendars, splicing telephone cables, scheduling round-robin tournaments, devising systematic methods for storing computer files, and generating pseudorandom numbers. … ►There are also applications of number theory in many diverse areas, including physics, biology, chemistry, communications, and art. …

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: 24.10 Arithmetic Properties

§24.10 Arithmetic Properties

… ►Here and elsewhere two rational numbers are congruent if the modulus divides the numerator of their difference. ►

§24.10(ii) Kummer Congruences

… ►

§24.10(iii) Voronoi’s Congruence

… ►

§24.10(iv) Factors

…

19: 24.14 Sums

§24.14 Sums

►

§24.14(i) Quadratic Recurrence Relations

… ►

24.14.2

\sum_{k=0}^{n}{n\choose k}B_{k}B_{n-k}=(1-n)B_{n}-nB_{n-1}.

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $k$ : integer and $n$ : integer
Permalink:: http://dlmf.nist.gov/24.14.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §24.14(i), §24.14 and Ch.24

… ►

§24.14(ii) Higher-Order Recurrence Relations

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

20: 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)}