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1: 27.20 Methods of Computation: Other Number-Theoretic Functions
The recursion formulas (27.14.6) and (27.14.7) can be used to calculate the partition function p ( n ) for n < N . …To compute a particular value p ( n ) it is better to use the Hardy-Ramanujan-Radematcher series (27.14.9). …
2: 26.9 Integer Partitions: Restricted Number and Part Size
p k ( n ) denotes the number of partitions of n into at most k parts. …
Table 26.9.1: Partitions p k ( n ) .
n k
p 1 ( n ) = 1 ,
3: 26.2 Basic Definitions
The total number of partitions of n is denoted by p ( n ) . …
Table 26.2.1: Partitions p ( n ) .
n p ( n ) n p ( n ) n p ( n )
4: 27.14 Unrestricted Partitions
§27.14(i) Partition Functions
as a generating function for the function p ( n ) defined in §27.14(i): …with p ( 0 ) = 1 . … where p ( k ) is defined to be 0 if k < 0 . … For example, p ( 10 ) = 42 , p ( 100 ) = 1905 69292 , and p ( 200 ) = 397 29990 29388 . …
5: 26.17 The Twelvefold Way
In this table ( k ) n is Pochhammer’s symbol, and S ( n , k ) and p k ( n ) are defined in §§26.8(i) and 26.9(i). …
Table 26.17.1: The twelvefold way.
elements of N elements of K f unrestricted f one-to-one f onto
unlabeled unlabeled p k ( n ) { 1 n k 0 n > k p k ( n ) - p k - 1 ( n )
6: 26.21 Tables
Abramowitz and Stegun (1964, Chapter 24) tabulates binomial coefficients ( m n ) for m up to 50 and n up to 25; extends Table 26.4.1 to n = 10 ; tabulates Stirling numbers of the first and second kinds, s ( n , k ) and S ( n , k ) , for n up to 25 and k up to n ; tabulates partitions p ( n ) and partitions into distinct parts p ( 𝒟 , n ) for n up to 500. …
7: 27.21 Tables
The partition function p ( n ) is tabulated in Gupta (1935, 1937), Watson (1937), and Gupta et al. (1958). …
8: 26.1 Special Notation
( m n ) binomial coefficient.
p ( n ) number of partitions of n .
p k ( n ) number of partitions of n into at most k parts.
9: 5.20 Physical Applications
Then the partition function (with β = 1 / ( k T ) ) is given by … and the partition function is given by
5.20.5 ψ n ( β ) = 1 ( 2 π ) n [ - π , π ] n e - β W d θ 1 d θ n = Γ ( 1 + 1 2 n β ) ( Γ ( 1 + 1 2 β ) ) - n .
Elementary Particles
Carlitz (1972) describes the partition function of dense hadronic matter in terms of a gamma function. …
10: 25.17 Physical Applications
The zeta function arises in the calculation of the partition function of ideal quantum gases (both Bose–Einstein and Fermi–Dirac cases), and it determines the critical gas temperature and density for the Bose–Einstein condensation phase transition in a dilute gas (Lifshitz and Pitaevskiĭ (1980)). …