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1: 26.9 Integer Partitions: Restricted Number and Part Size

… ►See Table 26.9.1. … ►

26.9.1 $$p_k\left(n\right)=p\left(n\right),$$ $k\ge n$.

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Symbols:

$p\left(n\right)$: total number of partitions of $n$ , $p_k\left(n\right)$: total number of partitions of $n$ into at most $k$ parts, $k$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.9.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §26.9(i), §26.9 and Ch.26

… ►

26.9.2 $$p_0\left(n\right)=0,$$ $n>0$,

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Symbols:

$p_k\left(n\right)$: total number of partitions of $n$ into at most $k$ parts and $n$: nonnegative integer

Referenced by:

§26.9(i)

Permalink:

http://dlmf.nist.gov/26.9.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §26.9(i), §26.9 and Ch.26

… ►

26.9.8 $$p_k\left(n\right)=p_k\left(n-k\right)+p_{k-1}\left(n\right);$$

ⓘ

Symbols:

$p_k\left(n\right)$: total number of partitions of $n$ into at most $k$ parts, $k$: nonnegative integer and $n$: nonnegative integer

Referenced by:

§26.10(iii)

Permalink:

http://dlmf.nist.gov/26.9.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §26.9(iii), §26.9 and Ch.26

… ►

26.9.10 $$p_k\left(n\right)\sim \frac{n^{k-1}}{k!(k-1)!}.$$

ⓘ

Symbols:

$\sim$: asymptotic equality, $!$: factorial (as in $n!$), $p_k\left(n\right)$: total number of partitions of $n$ into at most $k$ parts, $k$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.9.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §26.9(iv), §26.9 and Ch.26

2: 26.2 Basic Definitions

… ►The total number of partitions of $n$ is denoted by $p\left(n\right)$. …

3: 26.17 The Twelvefold Way

…

4: 27.14 Unrestricted Partitions

… ►

27.14.3 $$\frac{1}{f\left(x\right)}=\sum _{n=0}^{\mathrm{\infty }}p\left(n\right)x^n,$$

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Symbols:

$f\left(x\right)$: Euler’s reciprocal function, $p\left(n\right)$: total number of partitions of $n$ , $n$: positive integer and $x$: real number

A&S Ref:

24.2.1 I.B

Permalink:

http://dlmf.nist.gov/27.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §27.14(ii), §27.14 and Ch.27

… ►

27.14.6 $$p\left(n\right)=\sum _{k=1}^{\mathrm{\infty }}{(-1)}^{k+1}\left(p\left(n-\omega (k)\right)+p\left(n-\omega (-k)\right)\right)=p\left(n-1\right)+p\left(n-2\right)-p\left(n-5\right)-p\left(n-7\right)+\mathrm{\cdots },$$

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Symbols:

$p\left(n\right)$: total number of partitions of $n$ , $k$: positive integer, $n$: positive integer and $\omega (k)$: pentagonal numbers

A&S Ref:

24.2.1 II.A (in slightly different form)

Referenced by:

§27.20, §27.20

Permalink:

http://dlmf.nist.gov/27.14.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §27.14(ii), §27.14 and Ch.27

… ►

27.14.7 $$np\left(n\right)=\sum _{k=1}^n{\sigma }_1\left(k\right)p\left(n-k\right),$$

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Symbols:

${\sigma }_{\alpha }\left(n\right)$: sum of powers of divisors of a number, $p\left(n\right)$: total number of partitions of $n$ , $k$: positive integer and $n$: positive integer

A&S Ref:

24.2.1 II.A (in slightly different form) 24.2.1 II.A (in slightly different form)

Referenced by:

§27.20, §27.20, Erratum (V1.0.11) for Clarifications

Permalink:

http://dlmf.nist.gov/27.14.E7

Encodings:

pMML, png, TeX

Errata (effective with 1.0.11):

The erroneous ${\sigma }_1\left(n\right)$ in
$np\left(n\right)={\sum }_{k=1}^n{\sigma }_1\left(n\right)p\left(n-k\right)$
was replaced.

Reported 2015-08-17 by Howard Cohl and Hannah Cohen

See also:

Annotations for §27.14(ii), §27.14 and Ch.27

… ►

27.14.8 $$p\left(n\right)\sim \frac{{\mathrm{e}}^{K\sqrt{n}}}{4n\sqrt{3}},$$

ⓘ

Symbols:

$\sim$: asymptotic equality, $\mathrm{e}$: base of natural logarithm, $p\left(n\right)$: total number of partitions of $n$ , $n$: positive integer and $K$: change of variable

A&S Ref:

24.2.1 III

Permalink:

http://dlmf.nist.gov/27.14.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §27.14(iii), §27.14 and Ch.27

… ►

27.14.15 $$5\frac{{(f\left(x^5\right))}^5}{{(f\left(x\right))}^6}=\sum _{n=0}^{\mathrm{\infty }}p\left(5n+4\right)x^n$$

ⓘ

Symbols:

$f\left(x\right)$: Euler’s reciprocal function, $p\left(n\right)$: total number of partitions of $n$ , $n$: positive integer and $x$: real number

Permalink:

http://dlmf.nist.gov/27.14.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §27.14(v), §27.14 and Ch.27

…

5: DLMF Project News

error generating summary

6: 1.4 Calculus of One Variable

… ►

Functions of Bounded Variation

►With , the total variation of $f(x)$ on a finite or infinite interval $(a,b)$ is ►

1.4.33 $${𝒱}_{a,b}\left(f\right)=sup\sum _{j=1}^n\left|f(x_j)-f(x_{j-1})\right|,$$

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Defines:

$𝒱\left(f\right)$: total variation and ${𝒱}_{a,b}\left(f\right)$: total variation

Symbols:

$sup$: least upper bound (supremum), $j$: integer, $n$: nonnegative integer and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.4.E33

Encodings:

pMML, png, TeX

See also:

Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

… ►

1.4.34 $${𝒱}_{a,b}\left(f\right)={\int }_a^b\left|f^{\prime }(x)\right|dx,$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral, ${𝒱}_{a,b}\left(f\right)$: total variation and $\left|x\right|$: absolute value of $x$

Referenced by:

§1.4(v), §1.4(v), Erratum (V1.0.28) for Equation (1.4.34)

Permalink:

http://dlmf.nist.gov/1.4.E34

Encodings:

pMML, png, TeX

See also:

Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

… ►For further information on total variation see Olver (1997b, pp. 27–29). …

7: 31.14 General Fuchsian Equation

… ►With $a_1=0$ and $a_2=1$ the total number of free parameters is $3N-3$. …

8: 37.13 General Orthogonal Polynomials of $d$ Variables

… ►For $𝝂=({\nu }_1,\mathrm{\dots },{\nu }_d)\in {\mathbb{N}}_0^d$ and $𝐱=(x_1,\mathrm{\dots },x_d)\in {ℝ}^d$, the monomial ${𝐱}^{𝝂}=x_1^{{\nu }_1}\mathrm{\cdots }{x}_d^{{\nu }_d}$ has (total) degree $|𝝂|={\nu }_1+\mathrm{\cdots }+{\nu }_d$. …It is natural to use a graded order by ordering monomials by the total degree and choose a linear order (lexicographical order, for example) for monomials of the same total degree. …

9: 26.12 Plane Partitions

… ►A plane partition is totally symmetric if it is both symmetric and cyclically symmetric. The number of totally symmetric plane partitions in $B(r,r,r)$ is … ►The number of totally symmetric self-complementary plane partitions in $B(2r,2r,2r)$ is …

10: 25.10 Zeros

… ►Riemann developed a method for counting the total number $N(T)$ of zeros of $\zeta \left(s\right)$ in that portion of the critical strip with . …