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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: 22.9 Cyclic Identities

… ►These identities are cyclic in the sense that each of the indices

m, n

in the first product of, for example, the form

s_{m,p}^{(4)}s_{n,p}^{(4)}

are simultaneously permuted in the cyclic order:

m\to m+1\to m+2\to\cdots p\to 1\to 2\to\cdots m-1

;

n\to n+1\to n+2\to\cdots p\to 1\to 2\to\cdots n-1

. … …

4: 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)}

. …

5: 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

…

6: 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 …

7: 14.14 Continued Fractions

… ►provided that

x_{k+1}

and

y_{k}

do not vanish simultaneously for any

k=0,1,2,\dots

. …again provided

x_{k+1}

and

y_{k}

do not vanish simultaneously for any

k=0,1,2,\dots

8: 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. …

9: 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. …

10: 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

…