simultaneously permuted

♦ 11—20 of 43 matching pages ♦

(0.001 seconds)

11—20 of 43 matching pages

11: 26.17 The Twelvefold Way

§26.17 The Twelvefold Way

…

12: 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. …

13: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions

… ►

M_{2}

is the number of permutations of

\{1,2,\ldots,n\}

with

a_{1}

cycles of length 1,

a_{2}

cycles of length 2,

\dotsc

, and

a_{n}

cycles of length

n

: ►

26.4.7

M_{2}=\frac{n!}{1^{a_{1}}(a_{1}!)\,2^{a_{2}}(a_{2}!)\,\cdots\,n^{a_{n}}(a_{n}!% )}.

ⓘ

Symbols:: $!$ : factorial (as in $n!$ ), $n$ : nonnegative integer, $a_{n}$ : nonnegative integers and $M_{2}$ : number of permutations
Referenced by:: §26.13
Permalink:: http://dlmf.nist.gov/26.4.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §26.4(i), §26.4 and Ch.26

►(The empty set is considered to have one permutation consisting of no cycles.) …

14: 34.7 Basic Properties: $\mathit{9j}$ Symbol

… ►The

\mathit{9j}

symbol has symmetry properties with respect to permutation of columns, permutation of rows, and transposition of rows and columns; these relate 72 independent

\mathit{9j}

symbols. Even (cyclic) permutations of either columns or rows, as well as transpositions, leave the

\mathit{9j}

symbol unchanged. Odd permutations of columns or rows introduce a phase factor

(-1)^{R}

, where

R

is the sum of all arguments of the

\mathit{9j}

symbol. …

15: 19.21 Connection Formulas

… ►Let

y

z

, and

p

be positive and distinct, and permute

y

and

z

to ensure that

y

does not lie between

z

and

p

. … ►Because

R_{G}

is completely symmetric,

x, y, z

can be permuted on the right-hand side of (19.21.10) so that

(x-z)(y-z)\leq 0

if the variables are real, thereby avoiding cancellations when

R_{G}

is calculated from

R_{F}

and

R_{D}

(see §19.36(i)). …where both summations extend over the three cyclic permutations of

x, y, z

. … ►and

x, y, z

may be permuted. … ►For each value of

p

, permutation of

x, y, z

produces three values of

q

, one of which lies in the same region as

p

and two lie in the other region of the same type. …

16: 1.7 Inequalities

… ►Also

A

and

B

are constants that are not simultaneously zero. …

17: DLMF Project News

error generating summary

18: 19.23 Integral Representations

… ►With

l_{1},l_{2},l_{3}

denoting any permutation of

\sin\theta\cos\phi

\sin\theta\sin\phi

\cos\theta

, …

19: 1.2 Elementary Algebra

… ►

1.2.59

\det(\mathbf{A})=\sum_{\sigma\in\mathfrak{S}_{n}}\operatorname{sign}{\sigma}% \prod_{i=1}^{n}a_{i,\sigma(i)}.

ⓘ

Symbols:: $\det$ : determinant, $\in$ : element of, $\mathfrak{S}_{\NVar{n}}$ : set of permutations of $\{1,2,\ldots,n\}$ , $\operatorname{sign}\NVar{x}$ : sign of, $n$ : nonnegative integer and $\mathbf{A}$ : non-defective matrix
Permalink:: http://dlmf.nist.gov/1.2.E59
Encodings:: pMML, png, TeX
See also:: Annotations for §1.2(vi), §1.2(vi), §1.2 and Ch.1

►

\mathfrak{S}_{n}

is the set of all permutations of the set

\{1,2,...,n\}

. …

20: 1.3 Determinants, Linear Operators, and Spectral Expansions

… ►For self-adjoint

\mathbf{A}

and

\mathbf{B}

, if

[{\mathbf{A}},{\mathbf{B}}]=\boldsymbol{{0}}

, see (1.2.66), simultaneous eigenvectors of

\mathbf{A}

and

\mathbf{B}

always exist. …