inversion numbers
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1: 24.5 Recurrence Relations
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§24.5(iii) Inversion Formulas
…2: 27.5 Inversion Formulas
§27.5 Inversion Formulas
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27.5.2
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►Special cases of Möbius inversion pairs are:
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►Other types of Möbius inversion formulas include:
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3: 26.16 Multiset Permutations
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►The definitions of inversion number and major index can be extended to permutations of a multiset such as .
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4: 26.13 Permutations: Cycle Notation
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26.13.6
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►Given a permutation , the inversion number of , denoted , is the least number of adjacent transpositions required to represent .
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5: 26.15 Permutations: Matrix Notation
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►The inversion number of is a sum of products of pairs of entries in the matrix representation of :
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6: 26.14 Permutations: Order Notation
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►As an example, is an element of The inversion number is the number of pairs of elements for which the larger element precedes the smaller:
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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
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22.20.4
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9: 24.19 Methods of Computation
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