# permutations

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## 1—10 of 36 matching pages

##### 1: 26.13 Permutations: Cycle Notation

###### §26.13 Permutations: Cycle Notation

… ►The permutation … ►See §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

… ► ${\U0001d516}_{S}$ 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 ${\U0001d516}_{S}$ is the multinomial coefficient (§26.4) $\left(\genfrac{}{}{0.0pt}{}{{a}_{1}+{a}_{2}+\mathrm{\cdots}+{a}_{n}}{{a}_{1},{a}_{2},\mathrm{\dots},{a}_{n}}\right)$. … ►The definitions of inversion number and major index can be extended to permutations of a multiset such as $351322453154\in {\U0001d516}_{\{{1}^{2},{2}^{2},{3}^{3},{4}^{2},{5}^{3}\}}$. … ►
26.16.3
$$\sum _{\sigma \in {\U0001d516}_{S}}{q}^{maj(\sigma )}={\left[\genfrac{}{}{0.0pt}{}{{a}_{1}+{a}_{2}+\mathrm{\cdots}+{a}_{n}}{{a}_{1},{a}_{2},\mathrm{\dots},{a}_{n}}\right]}_{q}.$$

##### 3: 26.2 Basic Definitions

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###### 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 ${\U0001d516}_{n}$ with exactly $k+1$ weak excedances. … ►###### §26.14(iii) Identities

…##### 5: 26.15 Permutations: Matrix Notation

###### §26.15 Permutations: Matrix Notation

►The set ${\U0001d516}_{n}$ (§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

###### §19.15 Advantages of Symmetry

… ►The function ${R}_{-a}({b}_{1},{b}_{2},\mathrm{\dots},{b}_{n};{z}_{1},{z}_{2},\mathrm{\dots},{z}_{n})$ (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. …##### 7: Bille C. Carlson

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►The main theme of Carlson’s mathematical research has been to expose previously hidden permutation symmetries that can eliminate a set of transformations and thereby replace many formulas by a few.
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►In Symmetry in c, d, n of Jacobian elliptic functions (2004) he found a previously hidden symmetry in relations between Jacobian elliptic functions, which can now take a form that remains valid when the letters c, d, and n are permuted.
…In Permutation symmetry for theta functions (2011) he found an analogous hidden symmetry between theta functions.
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##### 8: 32.14 Combinatorics

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►Let ${S}_{N}$ be the group of permutations
$\bm{\pi}$ of the numbers $1,2,\mathrm{\dots},N$ (§26.2).
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32.14.1
$$\underset{N\to \mathrm{\infty}}{lim}\mathrm{Prob}\left(\frac{{\mathrm{\ell}}_{N}(\bm{\pi})-2\sqrt{N}}{{N}^{1/6}}\le s\right)=F(s),$$

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