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1: 26.13 Permutations: Cycle Notation

§26.13 Permutations: Cycle Notation

… ►is $\left(1,3,2,5,7\right)\left(4\right)\left(6,8\right)$ in cycle notation. Cycles of length one are fixed points. … ►For the example (26.13.2), this decomposition is given by $\left(1,3,2,5,7\right)\left(6,8\right)=\left(1,3\right)\left(2,3\right)\left(2,5\right)\left(5,7\right)\left(6,8\right).$ … ►Again, for the example (26.13.2) a minimal decomposition into adjacent transpositions is given by $\left(1,3,2,5,7\right)\left(6,8\right)=\left(2,3\right)\left(1,2\right)\left(4,5\right)\left(3,4\right)\left(2,3\right)\left(3,4\right)\left(4,5\right)\left(6,7\right)\left(5,6\right)\left(7,8\right)\left(6,7\right)$: $inv(\left(1,3,2,5,7\right)\left(6,8\right))=11$.

2: 26.2 Basic Definitions

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Cycle

►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 ${\sigma }^{\mathrm{\ell }-1}$. …If, for example, a permutation of the integers 1 through 6 is denoted by $256413$, then the cycles are $\left(1,2,5\right)$, $\left(3,6\right)$, and $\left(4\right)$. …

3: 21.7 Riemann Surfaces

… ►Removing the singularities of this curve gives rise to a two-dimensional connected manifold with a complex-analytic structure, that is, a Riemann surface. All compact Riemann surfaces can be obtained this way. … ►On this surface, we choose $2g$ cycles (that is, closed oriented curves, each with at most a finite number of singular points) $a_j$, $b_j$, $j=1,2,\mathrm{\dots },g$, such that their intersection indices satisfy … ►

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Figure 21.7.1: A basis of cycles for a genus 2 surface. Magnify

… ►Note that for the purposes of integrating these holomorphic differentials, all cycles on the surface are a linear combination of the cycles $a_j$, $b_j$, $j=1,2,\mathrm{\dots },g$. …

4: 21.1 Special Notation

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$g,h$	positive integers.
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$a\circ b$	intersection index of $a$ and $b$, two cycles lying on a closed surface. $a\circ b=0$ if $a$ and $b$ do not intersect. Otherwise $a\circ b$ gets an additive contribution from every intersection point. This contribution is $1$ if the basis of the tangent vectors of the $a$ and $b$ cycles (§21.7(i)) at the point of intersection is positively oriented; otherwise it is $-1$.
${\oint }_a\omega$	line integral of the differential $\omega$ over the cycle $a$.

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5: 4.13 Lambert $W$-Function

… ►For the definition of Stirling cycle numbers of the first kind $\left[\genfrac{}{}{0.0pt}{}{n}{k}\right]$ see (26.13.3). … ►

4.13.10 $$W_k\left(z\right)\sim {\xi }_k-\ln {\xi }_k+\sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^n}{{\xi }_k^n}\sum _{m=1}^n\left[\genfrac{}{}{0.0pt}{}{n}{n-m+1}\right]\frac{{\left(-\ln {\xi }_k\right)}^m}{m!},$$

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Symbols:

$W\left(z\right)$: Lambert $W$-function, $\left[\genfrac{}{}{0.0pt}{}{n}{k}\right]$: Stirling cycle number of the first kind, $\sim$: Poincaré asymptotic expansion, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $k$: integer, $m$: integer, $n$: integer and $z$: complex variable

Notes:

See Corless et al. (1996, p. 350) and Jeffrey et al. (1995, Theorem 2).

Referenced by:

§4.13, §4.13, §4.45(iii), Erratum (V1.1.7) for Expansion

Permalink:

http://dlmf.nist.gov/4.13.E10

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

Originally we gave only the first three terms of the infinite series.

See also:

Annotations for §4.13 and Ch.4

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4.13.11 $$W_{\pm 1}\left(x\mp 0\mathrm{i}\right)\sim -\eta -\ln \eta +\sum _{n=1}^{\mathrm{\infty }}\frac{1}{{\eta }^n}\sum _{m=1}^n\left[\genfrac{}{}{0.0pt}{}{n}{n-m+1}\right]\frac{{\left(-\ln \eta \right)}^m}{m!},$$

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Symbols:

$W\left(z\right)$: Lambert $W$-function, $\left[\genfrac{}{}{0.0pt}{}{n}{k}\right]$: Stirling cycle number of the first kind, $\sim$: Poincaré asymptotic expansion, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $m$: integer, $n$: integer, $x$: real variable and $\eta$

Notes:

See Corless et al. (1996, p. 350).

Referenced by:

§4.13, Erratum (V1.1.2) for Equation (4.13.11), Erratum (V1.1.7) for Expansion

Permalink:

http://dlmf.nist.gov/4.13.E11

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

Originally we gave only the first three terms of the infinite series.

See also:

Annotations for §4.13 and Ch.4

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6: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions

… ► $M_2$ is the number of permutations of $\{1,2,\mathrm{\dots },n\}$ with $a_1$ cycles of length 1, $a_2$ cycles of length 2, $\mathrm{\dots }$, and $a_n$ cycles of length $n$: …(The empty set is considered to have one permutation consisting of no cycles.) …

7: Bibliography G

… ►

General approach to few-cycle intense laser interactions with complex atoms. Phys. Rev. A 76, pp. 053411.

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External Links:

Document, Link

Referenced by:

§3.2(vi).

Encodings:

BibTeX

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8: 3.6 Linear Difference Equations

… ►If agreement is not within a prescribed tolerance the cycle is continued. …

9: 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. …