inversion numbers

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1: 24.5 Recurrence Relations

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§24.5(iii) Inversion Formulas

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2: 27.5 Inversion Formulas

§27.5 Inversion Formulas

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27.5.2 $$\sum _{d|n}\mu \left(d\right)=\lfloor \frac{1}{n}\rfloor ,$$

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Symbols:

$\mu \left(n\right)$: Möbius function, $\lfloor x\rfloor$: floor of $x$ , $d$: positive integer and $n$: positive integer

A&S Ref:

24.3.1 II.B (in slightly different form)

Referenced by:

§27.5

Permalink:

http://dlmf.nist.gov/27.5.E2

Encodings:

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See also:

Annotations for §27.5 and Ch.27

… ►Special cases of Möbius inversion pairs are: … ►Other types of Möbius inversion formulas include: … ►

3: 26.16 Multiset Permutations

… ►The definitions of inversion number and major index can be extended to permutations of a multiset such as $351322453154\in {𝔖}_{\{1^2,2^2,3^3,4^2,5^3\}}$. …

4: 26.13 Permutations: Cycle Notation

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26.13.6 $$\left(j,k\right)=\left(k-1,k\right)\left(k-2,k-1\right)\mathrm{\cdots }\left(j+1,j+2\right)\left(j,j+1\right)\left(j+1,j+2\right)\mathrm{\cdots }\left(k-1,k\right).$$

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Symbols:

$\left(S\right)$: cycle, $j$: nonnegative integer and $k$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.13.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §26.13 and Ch.26

… ►Given a permutation $\sigma \in {𝔖}_n$, the inversion number of $\sigma$, denoted $inv(\sigma )$, is the least number of adjacent transpositions required to represent $\sigma$. …

5: 26.15 Permutations: Matrix Notation

… ►The inversion number of $\sigma$ is a sum of products of pairs of entries in the matrix representation of $\sigma$: …

6: 26.14 Permutations: Order Notation

… ►As an example, $35247816$ is an element of ${𝔖}_8.$ The inversion number is the number of pairs of elements for which the larger element precedes the smaller: …

7: 27.17 Other Applications

§27.17 Other Applications

►Reed et al. (1990, pp. 458–470) describes a number-theoretic approach to Fourier analysis (called the arithmetic Fourier transform) that uses the Möbius inversion (27.5.7) to increase efficiency in computing coefficients of Fourier series. …

8: 37.17 Hermite Polynomials on ${ℝ}^d$

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37.17.20 $$\sum _{𝝂\in {\mathbb{N}}_0^d}{G}_{𝝂}(𝐱;𝐀)\frac{{𝐲}^{𝝂}}{𝝂!}={\mathrm{e}}^{2⟨𝐱,𝐲⟩-⟨{𝐀}^{-1}𝐲,𝐲⟩}.$$

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Symbols:

$\in$: element of, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $⟨𝐮,𝐯⟩$: inner product over vectors, ${𝐀}^{-1}$: matrix inverse, $\mathbb{N}$: set of all positive integers, $d$: positive integer, usually $\ge 2$ , $𝐱$: vector in $d$-dimensional Euclidean space, $𝐲$: vector in $d$-dimensional Euclidean space and ${\mathbb{N}}_0$: set of all nonnegative integers

Source:

Erdélyi et al. (1953b, (18) in §12.8)

Permalink:

http://dlmf.nist.gov/37.17.E20

Encodings:

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See also:

Annotations for §37.17(vi), §37.17(vi), §37.17, Part 37.III and Ch.37

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9: null

error generating summary

10: 22.20 Methods of Computation

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22.20.4 $${\varphi }_{n-1}=\frac{1}{2}\left({\varphi }_n+\arcsin \left(\frac{c_n}{a_n}\sin {\varphi }_n\right)\right),$$

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Symbols:

$\arcsin z$: arcsine function, $\sin z$: sine function, $a_n$: numbers, $c_n$: numbers, $n$: positive and ${\varphi }_N$: approximations

Referenced by:

§22.20(ii)

Permalink:

http://dlmf.nist.gov/22.20.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §22.20(ii), §22.20 and Ch.22

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