About the Project

inversion numbers

AdvancedHelp

(0.003 seconds)

1—10 of 49 matching pages

1: 24.5 Recurrence Relations
§24.5(iii) Inversion Formulas
2: 27.5 Inversion Formulas
§27.5 Inversion Formulas
27.5.2 d | n μ ( d ) = 1 n ,
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 𝔖 { 1 2 , 2 2 , 3 3 , 4 2 , 5 3 } . …
4: 26.13 Permutations: Cycle Notation
26.13.6 ( j , k ) = ( k 1 , k ) ( k 2 , k 1 ) ( j + 1 , j + 2 ) ( j , j + 1 ) ( j + 1 , j + 2 ) ( k 1 , k ) .
Given a permutation σ 𝔖 n , the inversion number of σ , denoted inv ( σ ) , is the least number of adjacent transpositions required to represent σ . …
5: 26.15 Permutations: Matrix Notation
The inversion number of σ is a sum of products of pairs of entries in the matrix representation of σ : …
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: 22.20 Methods of Computation
22.20.4 ϕ n 1 = 1 2 ( ϕ n + arcsin ( c n a n sin ϕ n ) ) ,
9: 24.19 Methods of Computation
  • Buhler et al. (1992) uses the expansion

    24.19.3 t 2 cosh t 1 = 2 n = 0 ( 2 n 1 ) B 2 n t 2 n ( 2 n ) ! ,

    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 d | n μ ( d ) f ( d ) = p | n ( 1 f ( p ) ) , n > 1 .
    Generating functions, Euler products, and Möbius inversion are used to evaluate many sums extended over divisors. …
    27.6.3 d | n | μ ( d ) | = 2 ν ( n ) ,