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11: 26.11 Integer Partitions: Compositions
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►For example, there are eight compositions of 4: , and .
… is the number of compositions of with no 1’s, where again .
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26.11.7
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12: About MathML
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►, built-in to the browser) support for MathML is growing, (see Browsers supporting MathML).
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Browsers supporting MathML
►The Firefox browser has traditionally had the strongest support for MathML and its native MathML is used by default. ►Recent enhancements to the WebKit engine now provide support for MathML Core. … ►Most modern browsers support ‘Web Fonts’, fonts that are effectively included with a web site. …13: 24.13 Integrals
14: 26.14 Permutations: Order Notation
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►The set (§26.13) can be viewed as the collection of all ordered lists of elements of : .
…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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►It is also equal to the number of permutations in with exactly weak excedances.
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26.14.5
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26.14.8
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15: 26.7 Set Partitions: Bell Numbers
16: 27.14 Unrestricted Partitions
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►For example, because there are exactly seven partitions of : .
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►with .
…where the exponents , , , , , , are the pentagonal numbers, defined by
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►Logarithmic differentiation of the generating function leads to another recursion:
…where is defined by (27.2.10) with .
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17: 24.14 Sums
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24.14.4
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24.14.7
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►Let denote a Hankel (or persymmetric) determinant, that is, an determinant with element in row and column for .
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24.14.11
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24.14.12
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18: 27.18 Methods of Computation: Primes
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►Two simple algorithms for proving primality require a knowledge of all or part of the factorization of , or both; see Crandall and Pomerance (2005, §§4.1–4.2).
These algorithms are used for testing primality of Mersenne numbers, , and Fermat numbers, .
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19: 3.11 Approximation Techniques
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►If is continuously differentiable on , then with
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► Moreover, the set of minimax approximations requires the calculation and storage of coefficients, whereas the corresponding set of Chebyshev-series approximations requires only coefficients.
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►With , the last equations give as the solution of a system of linear equations.
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►Here , , is a given set of distinct real points and .
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►(3.11.29) is a system of linear equations for the coefficients .
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