permutations

(0.001 seconds)

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

… ►

\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}.

ⓘ

Symbols:: $\in$ : element of, $\genfrac{[}{]}{0.0pt}{}{\NVar{a_{1}}+\NVar{a_{2}}+\dots+\NVar{a_{n}}}{\NVar{a_% {1}},\NVar{a_{2}},\ldots,\NVar{a_{n}}}_{\NVar{q}}$ : $q$ -multinomial coefficient, $\mathfrak{S}_{\NVar{n}}$ : set of permutations of $\{1,2,\ldots,n\}$ , $n$ : nonnegative integer, $a_{j}$ : numbers, $S$ : multiset and $q$ : parameter
Permalink:: http://dlmf.nist.gov/26.16.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §26.16 and Ch.26

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

\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. … ►

§26.14(iii) Identities

…

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

§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. …

7: Bille C. Carlson

… ►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. … ►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. …

8: 32.14 Combinatorics

… ►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),

ⓘ

Symbols:: $\boldsymbol{\pi}(m)$ : permutations, $\ell_{N}(\boldsymbol{\pi})$ : length and $F(s)$ : distribution function
Permalink:: http://dlmf.nist.gov/32.14.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §32.14 and Ch.32

…

9: 26.17 The Twelvefold Way

§26.17 The Twelvefold Way

…

10: 20.11 Generalizations and Analogs

… ►

§20.11(v) Permutation Symmetry

… ►The importance of these combined theta functions is that sets of twelve equations for the theta functions often can be replaced by corresponding sets of three equations of the combined theta functions, plus permutation symmetry. …