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1: 26.10 Integer Partitions: Other Restrictions
§26.10 Integer Partitions: Other Restrictions
►§26.10(i) Definitions
… ►§26.10(ii) Generating Functions
… ►§26.10(iii) Recurrence Relations
… ►§26.10(v) Limiting Form
…2: 26.11 Integer Partitions: Compositions
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►
denotes the number of compositions of , and is the number of compositions into exactly
parts.
is the number of compositions of with no 1’s, where again .
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►
26.11.1
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►
26.11.6
.
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3: 26.9 Integer Partitions: Restricted Number and Part Size
§26.9 Integer Partitions: Restricted Number and Part Size
►§26.9(i) Definitions
… ►§26.9(ii) Generating Functions
… ►§26.9(iii) Recurrence Relations
… ►§26.9(iv) Limiting Form
…4: 26.21 Tables
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►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.
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5: 26.1 Special Notation
6: 16.24 Physical Applications
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►They are also potentially useful for the solution of more complicated restricted lattice walk problems, and the 3D Ising model; see Barber and Ninham (1970, pp. 147–148).
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7: 15.7 Continued Fractions
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8: 26.15 Permutations: Matrix Notation
9: 10.44 Sums
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►If and the upper signs are taken, then the restriction on is unnecessary.
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►The restriction
is unnecessary when and is an integer.
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