# permutations

(0.001 seconds)

## 1—10 of 36 matching pages

##### 1: 26.13 Permutations: Cycle Notation
###### §26.13 Permutations: Cycle Notation
The permutationSee §26.8 for generating functions, recurrence relations, identities, and asymptotic approximations. A derangement is a permutation with no fixed points. … Every permutation is a product of transpositions. …
##### 2: 26.16 Multiset Permutations
###### §26.16 Multiset Permutations
$\mathfrak{S}_{S}$ denotes the set of permutations of $S$ for all distinct orderings of the $a_{1}+a_{2}+\cdots+a_{n}$ integers. The number of elements in $\mathfrak{S}_{S}$ is the multinomial coefficient (§26.4) $\genfrac{(}{)}{0.0pt}{}{a_{1}+a_{2}+\cdots+a_{n}}{a_{1},a_{2},\ldots,a_{n}}$. … The definitions of inversion number and major index can be extended to permutations of a multiset such as $351322453154\in\mathfrak{S}_{\{1^{2},2^{2},3^{3},4^{2},5^{3}\}}$. …
26.16.3 $\sum_{\sigma\in\mathfrak{S}_{S}}q^{\mathop{\mathrm{maj}}(\sigma)}=\genfrac{[}{% ]}{0.0pt}{}{a_{1}+a_{2}+\cdots+a_{n}}{a_{1},a_{2},\ldots,a_{n}}_{q}.$
##### 3: 26.2 Basic Definitions
###### Permutation
A permutation is a one-to-one and onto function from a non-empty set to itself. If the set consists of the integers 1 through $n$, a permutation $\sigma$ can be thought of as a rearrangement of these integers where the integer in position $j$ is $\sigma(j)$. Thus $231$ is the permutation $\sigma(1)=2$, $\sigma(2)=3$, $\sigma(3)=1$. … 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: 26.14 Permutations: Order Notation
###### §26.14 Permutations: Order Notation
The permutation $35247816$ has two descents: $52$ and $81$. … … It is also equal to the number of permutations in $\mathfrak{S}_{n}$ with exactly $k+1$ weak excedances. …
##### 5: 26.15 Permutations: Matrix Notation
###### §26.15 Permutations: Matrix Notation
The set $\mathfrak{S}_{n}$26.13) can be identified with the set of $n\times n$ matrices of 0’s and 1’s with exactly one 1 in each row and column. …The permutation $35247816$ corresponds to the matrix … The sign of the permutation $\sigma$ is the sign of the determinant of its matrix representation. … The number of permutations that avoid $B$ is …
##### 6: 19.15 Advantages of Symmetry
The function $R_{-a}\left(b_{1},b_{2},\dots,b_{n};z_{1},z_{2},\dots,z_{n}\right)$ (Carlson (1963)) reveals the full permutation symmetry that is partially hidden in $F_{D}$, and leads to symmetric standard integrals that simplify many aspects of theory, applications, and numerical computation. …
Let $S_{N}$ be the group of permutations $\boldsymbol{\pi}$ of the numbers $1,2,\dots,N$26.2). …
32.14.1 $\lim_{N\to\infty}\mathrm{Prob}\left(\frac{\ell_{N}(\boldsymbol{\pi})-2\sqrt{N}% }{N^{1/6}}\leq s\right)=F(s),$