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1: 26.10 Integer Partitions: Other Restrictions

§26.10 Integer Partitions: Other Restrictions

►

§26.10(i) Definitions

… ►

§26.10(ii) Generating Functions

… ►

§26.10(iii) Recurrence Relations

… ►

§26.10(v) Limiting Form

…

2: 26.11 Integer Partitions: Compositions

… ►

c\left(n\right)

denotes the number of compositions of

n

, and

c_{m}\left(n\right)

is the number of compositions into exactly

m

parts.

c\left(\in\!T,n\right)

is the number of compositions of

n

with no 1’s, where again

T=\{2,3,4,\ldots\}

. … ►

26.11.1

c\left(0\right)=c\left(\in\!T,0\right)=1.

ⓘ

Symbols:: $\in$ : element of, $c\left(\NVar{n}\right)$ : number of compositions of $n$ , $c\left(\NVar{\mathrm{condition}},\NVar{n}\right)$ : restricted number of compositions of $n$ and $T$ : set
Permalink:: http://dlmf.nist.gov/26.11.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §26.11 and Ch.26

… ►

26.11.6

c\left(\in\!T,n\right)=F_{n-1},

n\geq 1

ⓘ

Symbols:: $\in$ : element of, $c\left(\NVar{\mathrm{condition}},\NVar{n}\right)$ : restricted number of compositions of $n$ , $n$ : nonnegative integer, $T$ : set and $F_{n}$ : Fibonacci numbers
Permalink:: http://dlmf.nist.gov/26.11.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §26.11 and Ch.26

…

3: 26.9 Integer Partitions: Restricted Number and Part Size

§26.9 Integer Partitions: Restricted Number and Part Size

►

§26.9(i) Definitions

… ►

§26.9(ii) Generating Functions

… ►

§26.9(iii) Recurrence Relations

… ►

§26.9(iv) Limiting Form

…

4: 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. …

5: 26.1 Special Notation

… ► ►►

$x$	real variable.
…

…

6: 16.24 Physical Applications

… ►They are also potentially useful for the solution of more complicated restricted lattice walk problems, and the 3D Ising model; see Barber and Ninham (1970, pp. 147–148). …

7: 15.7 Continued Fractions

…

8: 26.15 Permutations: Matrix Notation

… ►where the sum is over

1\leq g<k\leq n

and

n\geq h>\ell\geq 1

. … ►A permutation with restricted position specifies a subset

B\subseteq\{1,2,\ldots,n\}\times\{1,2,\ldots,n\}

. …

9: 10.44 Sums

… ►If

\mathscr{Z}=I

and the upper signs are taken, then the restriction on

\lambda

is unnecessary. … ►The restriction

|v|<|u|

is unnecessary when

\mathscr{Z}=I

and

\nu

is an integer. …

10: 14.25 Integral Representations

… ►For corresponding contour integrals, with less restrictions on

\mu

and

\nu

, see Olver (1997b, pp. 174–179), and for further integral representations see Magnus et al. (1966, §4.6.1).