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11: 26.16 Multiset Permutations
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►Let be the multiset that has copies of , .
denotes the set of permutations of for all distinct orderings of the integers.
The number of elements in is the multinomial coefficient (§26.4) .
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-multinomial coefficient is defined in terms of Gaussian polynomials (§26.9(ii)) by
…and again with we have
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12: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions
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is the number of ways of placing distinct objects into labeled boxes so that there are objects in the th box.
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►These are given by the following equations in which are nonnegative integers such that
… is the number of permutations of with cycles of length 1, cycles of length 2, , and cycles of length :
…For each all possible values of are covered.
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►where the summation is over all nonnegative integers such that .
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13: 5.10 Continued Fractions
14: 18.8 Differential Equations
15: 25.20 Approximations
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Cody et al. (1971) gives rational approximations for in the form of quotients of polynomials or quotients of Chebyshev series. The ranges covered are , , , . Precision is varied, with a maximum of 20S.
Piessens and Branders (1972) gives the coefficients of the Chebyshev-series expansions of and , , for (23D).
16: 21.1 Special Notation
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►The function is also commonly used; see, for example, Belokolos et al. (1994, §2.5), Dubrovin (1981), and Fay (1973, Chapter 1).
positive integers. | |
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Transpose of . | |
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set of all elements of the form “”. | |
set of all elements of , modulo elements of . Thus two elements of are equivalent if they are both in and their difference is in . (For an example see §20.12(ii).) | |
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17: 24.2 Definitions and Generating Functions
18: 10.75 Tables
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Makinouchi (1966) tabulates all values of and in the interval , with at least 29S. These are for , 10, 20; , ; with and , except for .
Abramowitz and Stegun (1964, Chapter 11) tabulates , , , 10D; , , , 8D.
Leung and Ghaderpanah (1979), tabulates all zeros of the principal value of , for , 29S.
Abramowitz and Stegun (1964, Chapter 11) tabulates , , , 7D; , , , 6D.
19: 26.12 Plane Partitions
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►The number of self-complementary plane partitions in is
…in it is
…in it is
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►The notation denotes the sum over all plane partitions contained in , and denotes the number of elements in .
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►where is the sum of the squares of the divisors of .
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20: Bibliography
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Asymptotic expansions of spheroidal wave functions.
J. Math. Phys. Mass. Inst. Tech. 28, pp. 195–199.
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On the zeros of confluent hypergeometric functions. III. Characterization by means of nonlinear equations.
Lett. Nuovo Cimento (2) 29 (11), pp. 353–358.
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Transformations of the ranks and algebraic solutions of the sixth Painlevé equation.
Comm. Math. Phys. 228 (1), pp. 151–176.
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Special value of the hypergeometric function and connection formulae among asymptotic expansions.
J. Indian Math. Soc. (N.S.) 51, pp. 161–221.
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Normal forms of functions in the neighborhood of degenerate critical points.
Uspehi Mat. Nauk 29 (2(176)), pp. 11–49 (Russian).
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