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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 ${\left(1,3,2,5,7\right)}{\left(6,8\right)}={\left(2,3\right)}\*{\left(1,2% \right)}\*{\left(4,5\right)}{\left(3,4\right)}{\left(2,3\right)}{\left(3,4% \right)}{\left(4,5\right)}{\left(6,7\right)}{\left(5,6\right)}{\left(7,8\right% )}\*{\left(6,7\right)}$: $\mathop{\mathrm{inv}}({\left(1,3,2,5,7\right)}{\left(6,8\right)})=11$.
2: 26.2 Basic Definitions
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 $\ell=\ell(j,k)$ such that $j=\sigma^{\ell}(k)$, where $\sigma^{1}=\sigma$ and $\sigma^{\ell}$ is the composition of $\sigma$ with $\sigma^{\ell-1}$. …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) $a_{j}$, $b_{j}$, $j=1,2,\dots,g$, such that their intersection indices satisfy … Note that for the purposes of integrating these holomorphic differentials, all cycles on the surface are a linear combination of the cycles $a_{j}$, $b_{j}$, $j=1,2,\dots,g$. …
4: 21.1 Special Notation
 $g,h$ positive integers. … 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 $-1$. line integral of the differential $\omega$ over the cycle $a$.
5: 4.13 Lambert $W$-Function
For the definition of Stirling cycle numbers of the first kind $\genfrac{[}{]}{0.0pt}{}{n}{k}$ see (26.13.3). …
6: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions
$M_{2}$ is the number of permutations of $\{1,2,\ldots,n\}$ with $a_{1}$ cycles of length 1, $a_{2}$ cycles of length 2, $\dotsc$, and $a_{n}$ cycles of length $n$: …(The empty set is considered to have one permutation consisting of no cycles.) …
7: Bibliography G
• X. Guan, O. Zatsarinny, K. Bartschat, B. I. Schneider, J. Feist, and C. J. Noble (2007) General approach to few-cycle intense laser interactions with complex atoms. Phys. Rev. A 76, pp. 053411.
• 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
$s\left(n,k\right)$ denotes the Stirling number of the first kind: $(-1)^{n-k}$ times the number of permutations of $\{1,2,\ldots,n\}$ with exactly $k$ cycles. …