# cycle notation

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##### 1: 26.13 Permutations: Cycle Notation
###### §26.13 Permutations: CycleNotation
In cycle notation, the elements in each cycle are put inside parentheses, ordered so that $\sigma(j)$ immediately follows $j$ or, if $j$ is the last listed element of the cycle, then $\sigma(j)$ is the first element of the cycle. … is ${\left(1,3,2,5,7\right)}{\left(4\right)}{\left(6,8\right)}$ in cycle notation. …They are often dropped from the cycle notation. … 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: 21.1 Special Notation
###### §21.1 Special Notation
(For other notation see Notation for the Special Functions.)
 $g,h$ 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 $a$.
Uppercase boldface letters are $g\times g$ real or complex matrices. …
##### 3: 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 ${\left(4\right)}$. …
##### 4: 4.13 Lambert $W$-Function
###### §4.13 Lambert $W$-Function
Alternative notations are $\operatorname{Wp}\left(x\right)$ for $W_{0}\left(x\right)$, $\operatorname{Wm}\left(x\right)$ for $W_{-1}\left(x+0\mathrm{i}\right)$, both previously used in this section, the Wright $\omega$-function $\omega\left(z\right)=W\left({\mathrm{e}}^{z}\right)$, which is single-valued, satisfies … For the definition of Stirling cycle numbers of the first kind $\genfrac{[}{]}{0.0pt}{}{n}{k}$ see (26.13.3). …
##### 5: 32.1 Special Notation
###### §32.1 Special Notation
(For other notation see Notation for the Special Functions.) …
##### 6: 6.1 Special Notation
###### §6.1 Special Notation
(For other notation see Notation for the Special Functions.) … Unless otherwise noted, primes indicate derivatives with respect to the argument. …
##### 7: 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 $a_{j}$, $b_{j}$, $j=1,2,\dots,g$. …
##### 8: 24.1 Special Notation
###### §24.1 Special Notation
(For other notation see Notation for the Special Functions.) …
###### Bernoulli Numbers and Polynomials
Among various older notations, the most common one is …
##### 9: 4.1 Special Notation
###### §4.1 Special Notation
(For other notation see Notation for the Special Functions.) … The main purpose of the present chapter is to extend these definitions and properties to complex arguments $z$. …
##### 10: 5.1 Special Notation
###### §5.1 Special Notation
(For other notation see Notation for the Special Functions.) … The notation $\Gamma\left(z\right)$ is due to Legendre. Alternative notations for this function are: $\Pi(z-1)$ (Gauss) and $(z-1)!$. Alternative notations for the psi function are: $\Psi(z-1)$ (Gauss) Jahnke and Emde (1945); $\Psi(z)$ Davis (1933); $\mathsf{F}(z-1)$ Pairman (1919). …