高仿花旗银行报表【言正 微aptao168】45S

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1: 26.16 Multiset Permutations

… ►Let $S=\{1^{a_1},2^{a_2},\mathrm{\dots },n^{a_n}\}$ be the multiset that has $a_j$ copies of $j$, $1\le j\le n$. ${𝔖}_S$ denotes the set of permutations of $S$ for all distinct orderings of the $a_1+a_2+\mathrm{\cdots }+a_n$ integers. The number of elements in ${𝔖}_S$ is the multinomial coefficient (§26.4) $\left(\genfrac{}{}{0.0pt}{}{a_1+a_2+\mathrm{\cdots }+a_n}{a_1,a_2,\mathrm{\dots },a_n}\right)$. Additional information can be found in Andrews (1976, pp. 39–45). … ►and again with $S=\{1^{a_1},2^{a_2},\mathrm{\dots },n^{a_n}\}$ we have …

2: 26.8 Set Partitions: Stirling Numbers

… ► $s(n,k)$ denotes the Stirling number of the first kind: ${(-1)}^{n-k}$ times the number of permutations of $\{1,2,\mathrm{\dots },n\}$ with exactly $k$ cycles. … … ► $S(n,k)$ denotes the Stirling number of the second kind: the number of partitions of $\{1,2,\mathrm{\dots },n\}$ into exactly $k$ nonempty subsets. … ►Let $A$ and $B$ be the $n\times n$ matrices with $(j,k)$th elements $s(j,k)$, and $S(j,k)$, respectively. … ►For asymptotic approximations for $s(n+1,k+1)$ and $S(n,k)$ that apply uniformly for $1\le k\le n$ as $n\to \mathrm{\infty }$ see Temme (1993) and Temme (2015, Chapter 34). …

3: 26.2 Basic Definitions

… ►Given a finite set $S$ with permutation $\sigma$, a cycle is an ordered equivalence class of elements of $S$ where $j$ is equivalent to $k$ if there exists an $\mathrm{\ell }=\mathrm{\ell }(j,k)$ such that $j={\sigma }^{\mathrm{\ell }}(k)$, where ${\sigma }^1=\sigma$ and ${\sigma }^{\mathrm{\ell }}$ is the composition of $\sigma$ with ${\sigma }^{\mathrm{\ell }-1}$. … ►A partition of a set $S$ is an unordered collection of pairwise disjoint nonempty sets whose union is $S$. … ►

Table 26.2.1: Partitions $p\left(n\right)$.

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$n$	$p\left(n\right)$	$n$	$p\left(n\right)$	$n$	$p\left(n\right)$
…
11	56	28	3718	45	89134
…

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4: 21.1 Special Notation

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$g,h$	positive integers.
…
$S^g$	set of $g$-dimensional vectors with elements in $S$.
$|S|$	number of elements of the set $S$.
$S_1{S}_2$	set of all elements of the form “$\text{element of }{S}_1\times \text{element of }{S}_2$”.
$S_1/S_2$	set of all elements of $S_1$, modulo elements of $S_2$. Thus two elements of $S_1/S_2$ are equivalent if they are both in $S_1$ and their difference is in $S_2$. (For an example see §20.12(ii).)
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5: 26.1 Special Notation

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$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$	binomial coefficient.
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$s(n,k)$	Stirling numbers of the first kind.
$S(n,k)$	Stirling numbers of the second kind.

… ►Other notations for $s(n,k)$, 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)), $\left(\genfrac{}{}{0.0pt}{}{n-1}{k-1}\right)B_{n-k}^{(n)}$ (Milne-Thomson (1933)), ${(-1)}^{n-k}{S}_1(n-1,n-k)$ (Carlitz (1960), Gould (1960)), ${(-1)}^{n-k}\left[\genfrac{}{}{0pt}{}{n}{k}\right]$ (Knuth (1992), Graham et al. (1994), Rosen et al. (2000)). ►Other notations for $S(n,k)$, the Stirling numbers of the second kind, include ${𝒮}_n^{(k)}$ (Fort (1948)), ${𝔖}_n^k$ (Jordan (1939)), ${\sigma }_n^k$ (Moser and Wyman (1958b)), $\left(\genfrac{}{}{0.0pt}{}{n}{k}\right)B_{n-k}^{(-k)}$ (Milne-Thomson (1933)), $S_2(k,n-k)$ (Carlitz (1960), Gould (1960)), $\left\{\genfrac{}{}{0pt}{}{n}{k}\right\}$ (Knuth (1992), Graham et al. (1994), Rosen et al. (2000)), and also an unconventional symbol in Abramowitz and Stegun (1964, Chapter 24).

6: 3.9 Acceleration of Convergence

… ►If $S={\sum }_{k=0}^{\mathrm{\infty }}{(-1)}^k{a}_k$ is a convergent series, then ►

3.9.2 $$S=\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k{2}^{-k-1}{\mathrm{\Delta }}^k{a}_0,$$

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Symbols:

$\mathrm{\Delta }$: forward difference operator and $S$: a convergent series

Permalink:

http://dlmf.nist.gov/3.9.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §3.9(ii), §3.9 and Ch.3

… ►

Table 3.9.1: Shanks’ transformation for $s_n={\sum }_{j=1}^n{(-1)}^{j+1}{j}^{-2}$.

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$n$	$t_{n,2}$	$t_{n,4}$	$t_{n,6}$	$t_{n,8}$	$t_{n,10}$
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1	0.82692 30769 23	0.82259 02017 65	0.82247 05346 57	0.82246 71342 06	0.82246 70363 45
2	0.82111 11111 11	0.82243 44785 14	0.82246 61821 45	0.82246 70102 48	0.82246 70327 79
…

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7: 30.11 Radial Spheroidal Wave Functions

… ►

$S_n^{m(3)}(z,\gamma )=S_n^{m(1)}(z,\gamma )+\mathrm{i}{S}_n^{m(2)}(z,\gamma ),$

►

$S_n^{m(4)}(z,\gamma )=S_n^{m(1)}(z,\gamma )-\mathrm{i}{S}_n^{m(2)}(z,\gamma ).$

… ►

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Figure 30.11.1: $S_n^{0(1)}(x,2)$, $n=0,1$, $1\le x\le 10$. Magnify

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Figure 30.11.2: $S_n^{0(1)}(\mathrm{i}y,2\mathrm{i})$, $n=0,1$, $0\le y\le 10$. Magnify

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►►

Figure 30.11.3: $S_n^{1(1)}(x,2)$, $n=1,2$, $1\le x\le 10$. Magnify

…

8: 26.17 The Twelvefold Way

… ►In this table ${\left(k\right)}_n$ is Pochhammer’s symbol, and $S(n,k)$ and $p_k\left(n\right)$ are defined in §§26.8(i) and 26.9(i). … ►

Table 26.17.1: The twelvefold way.

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elements of $N$	elements of $K$	$f$ unrestricted	$f$ one-to-one	$f$ onto
labeled	labeled	$k^n$	${\left(k-n+1\right)}_n$	$k!S(n,k)$
…
labeled	unlabeled	$\begin{array}{c}S(n,1)+S(n,2)\\ +\mathrm{\cdots }+S(n,k)\end{array}$		$S(n,k)$
…

►

9: 19.29 Reduction of General Elliptic Integrals

… ►where … ►The first choice gives a formula that includes the 18+9+18 = 45 formulas in Gradshteyn and Ryzhik (2000, 3.133, 3.156, 3.158), and the second choice includes the 8+8+8+12 = 36 formulas in Gradshteyn and Ryzhik (2000, 3.151, 3.149, 3.137, 3.157) (after setting $x^2=t$ in some cases). … ►where ►

${(x-y)}^{2}U=S_1{S}_2,$

… ►

$S_1={(X_1+Y_1)}^2,$

…

10: 15.13 Zeros

… ►

15.13.1

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Symbols:

$\lfloor x\rfloor$: floor of $x$ , $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter, $N(a,b,c)$: number of zeros and $S$

Permalink:

http://dlmf.nist.gov/15.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §15.13 and Ch.15

►where $S=\mathrm{sign}\left(\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(b\right)\mathrm{\Gamma }\left(c-a\right)\mathrm{\Gamma }\left(c-b\right)\right)$. …