# inversion numbers

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##### 2: 27.5 Inversion Formulas
###### §27.5 Inversion Formulas
27.5.2 $\sum_{d\mathbin{|}n}\mu\left(d\right)=\left\lfloor\frac{1}{n}\right\rfloor,$
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 $351322453154\in\mathfrak{S}_{\{1^{2},2^{2},3^{3},4^{2},5^{3}\}}$. …
##### 4: 26.13 Permutations: Cycle Notation
26.13.6 ${\left(j,k\right)}={\left(k-1,k\right)}{\left(k-2,k-1\right)}\cdots{\left(j+1,% j+2\right)}\*{\left(j,j+1\right)}{\left(j+1,j+2\right)}\cdots{\left(k-1,k% \right)}.$
Given a permutation $\sigma\in\mathfrak{S}_{n}$, the inversion number of $\sigma$, denoted $\mathop{\mathrm{inv}}(\sigma)$, 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 $\mathfrak{S}_{8}.$ 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. …
##### 9: 24.19 Methods of Computation
• Buhler et al. (1992) uses the expansion

and computes inverses modulo $p$ of the left-hand side. Multisectioning techniques are applied in implementations. See also Crandall (1996, pp. 116–120).

• ##### 10: 27.6 Divisor Sums
###### §27.6 Divisor Sums
Sums of number-theoretic functions extended over divisors are of special interest. …
27.6.2 $\sum_{d\mathbin{|}n}\mu\left(d\right)f(d)=\prod_{p\mathbin{|}n}(1-f(p)),$ $n>1$.
Generating functions, Euler products, and Möbius inversion are used to evaluate many sums extended over divisors. …