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11: 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:
…Equivalently, this is the sum over of the number of integers less than that lie in positions to the right of the th position:
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►The Eulerian number, denoted , is the number of permutations in with exactly descents.
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§26.14(iii) Identities
…12: 26.7 Set Partitions: Bell Numbers
§26.7 Set Partitions: Bell Numbers
►§26.7(i) Definitions
► is the number of partitions of . … ►§26.7(ii) Generating Function
… ►§26.7(iii) Recurrence Relation
…13: 24.19 Methods of Computation
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§24.19(i) Bernoulli and Euler Numbers and Polynomials
… ►A similar method can be used for the Euler numbers based on (4.19.5). … ►§24.19(ii) Values of Modulo
►For number-theoretic applications it is important to compute for ; in particular to find the irregular pairs for which . … ►14: 27.2 Functions
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►where are the distinct prime factors of , each exponent is positive, and is the number of distinct primes dividing .
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►(See Gauss (1863, Band II, pp. 437–477) and Legendre (1808, p. 394).)
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►The
numbers
are relatively prime to and distinct (mod ).
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§27.2(ii) Tables
…15: 24.10 Arithmetic Properties
§24.10 Arithmetic Properties
… ►The denominator of is the product of all these primes . … ►§24.10(ii) Kummer Congruences
… ►§24.10(iii) Voronoi’s Congruence
… ►§24.10(iv) Factors
…16: 26.8 Set Partitions: Stirling Numbers
§26.8 Set Partitions: Stirling Numbers
►§26.8(i) Definitions
► denotes the Stirling number of the first kind: times the number of permutations of with exactly cycles. … ► denotes the Stirling number of the second kind: the number of partitions of into exactly nonempty subsets. … ►§26.8(vi) Relations to Bernoulli Numbers
…17: 24.14 Sums
§24.14 Sums
►§24.14(i) Quadratic Recurrence Relations
… ►§24.14(ii) Higher-Order Recurrence Relations
►In the following two identities, valid for , the sums are taken over all nonnegative integers with . … ►For other sums involving Bernoulli and Euler numbers and polynomials see Hansen (1975, pp. 331–347) and Prudnikov et al. (1990, pp. 383–386).18: 26.21 Tables
§26.21 Tables
►Abramowitz and Stegun (1964, Chapter 24) tabulates binomial coefficients for up to 50 and up to 25; extends Table 26.4.1 to ; tabulates Stirling numbers of the first and second kinds, and , for up to 25 and up to ; tabulates partitions and partitions into distinct parts for up to 500. ►Andrews (1976) contains tables of the number of unrestricted partitions, partitions into odd parts, partitions into parts , partitions into parts , and unrestricted plane partitions up to 100. It also contains a table of Gaussian polynomials up to . ►Goldberg et al. (1976) contains tables of binomial coefficients to and Stirling numbers to .19: 26.13 Permutations: Cycle Notation
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►The number of elements of with cycle type is given by (26.4.7).
►The Stirling cycle numbers of the first kind, denoted by , count the number of permutations of with exactly cycles.
They are related to Stirling numbers of the first kind by
…See §26.8 for generating functions, recurrence relations, identities, and asymptotic approximations.
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►The derangement number, , is the number of elements of with no fixed points:
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20: 26.1 Special Notation
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►Other notations for , the Stirling numbers of the first kind, include (Abramowitz and Stegun (1964, Chapter 24), Fort (1948)), (Jordan (1939), Moser and Wyman (1958a)), (Milne-Thomson (1933)), (Carlitz (1960), Gould (1960)), (Knuth (1992), Graham et al. (1994), Rosen et al. (2000)).
►Other notations for , the Stirling numbers of the second kind, include (Fort (1948)), (Jordan (1939)), (Moser and Wyman (1958b)), (Milne-Thomson (1933)), (Carlitz (1960), Gould (1960)), (Knuth (1992), Graham et al. (1994), Rosen et al. (2000)), and also an unconventional symbol in Abramowitz and Stegun (1964, Chapter 24).
binomial coefficient. | |
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Eulerian number. | |
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Bell number. | |
Catalan number. | |
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