restricted%20integer%20partitions

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1: 28.2 Definitions and Basic Properties

… ►The general solution of (28.2.16) is

\nu=\pm\widehat{\nu}+2n

, where

n\in\mathbb{Z}

. … ►

28.2.19

{qc_{2n+2}-\left(a-(\nu+2n)^{2}\right)c_{2n}+qc_{2n-2}=0,}

n\in\mathbb{Z}

ⓘ

Symbols:: $\in$ : element of, $\mathbb{Z}$ : set of all integers, $q=h^{2}$ : parameter, $n$ : integer, $\nu$ : complex parameter, $a$ : parameter and $c_{2n}$ : coefficients
A&S Ref:: 20.3.12 20.3.13
Referenced by:: §28.2(iv)
Permalink:: http://dlmf.nist.gov/28.2.E19
Encodings:: pMML, png, TeX
See also:: Annotations for §28.2(iv), §28.2 and Ch.28

… ►

28.2.23

a_{n}\left(0\right)=n^{2},

n=0,1,2,\dots

ⓘ

Symbols:: $a_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation and $n$ : integer
Permalink:: http://dlmf.nist.gov/28.2.E23
Encodings:: pMML, png, TeX
See also:: Annotations for §28.2(v), §28.2 and Ch.28

►

28.2.24

b_{n}\left(0\right)=n^{2},

n=1,2,3,\dots

ⓘ

Symbols:: $b_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation and $n$ : integer
Permalink:: http://dlmf.nist.gov/28.2.E24
Encodings:: pMML, png, TeX
See also:: Annotations for §28.2(v), §28.2 and Ch.28

… ►

§28.2(vi) Eigenfunctions

…

2: 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

…

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: 26.11 Integer Partitions: Compositions

§26.11 Integer Partitions: Compositions

►A composition is an integer partition in which order is taken into account. …

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\}

. The integer 0 is considered to have one composition consisting of no parts: …

5: 26.2 Basic Definitions

… ►

Partition

… ►As an example,

\{1,3,4\}

\{2,6\}

\{5\}

is a partition of

\{1,2,3,4,5,6\}

. ►A partition of a nonnegative integer

n

is an unordered collection of positive integers whose sum is

n

. …The total number of partitions of

n

is denoted by

p\left(n\right)

. … ►The integers whose sum is

n

are referred to as the parts in the partition. …

6: 26.12 Plane Partitions

§26.12 Plane Partitions

►

§26.12(i) Definitions

►A plane partition,

\pi

, of a positive integer

n

, is a partition of

n

in which the parts have been arranged in a 2-dimensional array that is weakly decreasing (nonincreasing) across rows and down columns. … ► … ►The plane partition in Figure 26.12.1 is an example of a cyclically symmetric plane partition. …

7: 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. …

8: 20 Theta Functions

Chapter 20 Theta Functions

…

9: 26.1 Special Notation

… ► ►►►►

$x$	real variable.
…
$\lambda$	integer partition.
$\pi$	plane partition.
…

… ► ►►►►

$\genfrac{(}{)}{0.0pt}{}{m}{n}$	binomial coefficient.
…
$p\left(n\right)$	number of partitions of $n$ .
…
$\operatorname{pp}\left(n\right)$	number of plane partitions of $n$ .
…

…

10: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions

§26.4 Lattice Paths: Multinomial Coefficients and Set Partitions

… ►Table 26.4.1 gives numerical values of multinomials and partitions

\lambda,M_{1},M_{2},M_{3}

for

1\leq m\leq n\leq 5

. These are given by the following equations in which

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

are nonnegative integers such that …

\lambda

is a partition of

n

: …

M_{3}

is the number of set partitions of

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

with

a_{1}

subsets of size 1,

a_{2}

subsets of size 2,

\dotsc

, and

a_{n}

subsets of size

n

: …