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Stirling numbers (first and second kinds)

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1: 26.8 Set Partitions: Stirling Numbers
Table 26.8.1: Stirling numbers of the first kind s ( n , k ) .
n k
s ( n , 0 ) = 0 ,
S ( n , 0 ) = 0 ,
S ( n , 1 ) = 1 ,
2: 26.1 Special Notation
x

real variable.

( m n )

binomial coefficient.

s ( n , k )

Stirling numbers of the first kind.

S ( n , k )

Stirling numbers of the second kind.

Other notations for s ( n , k ) , the Stirling numbers of the first kind, include S n ( k ) (Abramowitz and Stegun (1964, Chapter 24), Fort (1948)), S n k (Jordan (1939), Moser and Wyman (1958a)), ( n - 1 k - 1 ) B n - k ( n ) (Milne-Thomson (1933)), ( - 1 ) n - k S 1 ( n - 1 , n - k ) (Carlitz (1960), Gould (1960)), ( - 1 ) n - k [ n k ] (Knuth (1992), Graham et al. (1994), Rosen et al. (2000)). Other notations for S ( n , k ) , the Stirling numbers of the second kind, include 𝒮 n ( k ) (Fort (1948)), 𝔖 n k (Jordan (1939)), σ n k (Moser and Wyman (1958b)), ( n k ) B n - k ( - k ) (Milne-Thomson (1933)), S 2 ( k , n - k ) (Carlitz (1960), Gould (1960)), { n k } (Knuth (1992), Graham et al. (1994), Rosen et al. (2000)), and also an unconventional symbol in Abramowitz and Stegun (1964, Chapter 24).
3: 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
labeled labeled k n ( k - n + 1 ) n k ! S ( n , k )
labeled unlabeled S ( n , 1 ) + S ( n , 2 ) + + S ( n , k ) { 1 n k 0 n > k S ( n , k )
4: 26.21 Tables
§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. … It also contains a table of Gaussian polynomials up to [ 12 6 ] q . …
5: 24.15 Related Sequences of Numbers
The Stirling numbers of the first kind s ( n , m ) , and the second kind S ( n , m ) , are as defined in §26.8(i). …
24.15.7 B n = k = 0 n ( - 1 ) k ( n + 1 k + 1 ) S ( n + k , k ) / ( n + k k ) ,
24.15.8 k = 0 n ( - 1 ) n + k s ( n + 1 , k + 1 ) B k = n ! n + 1 .
24.15.9 p B n n S ( p - 1 + n , p - 1 ) ( mod p 2 ) , 1 n p - 2 ,
24.15.10 2 n - 1 4 n p 2 B 2 n S ( p + 2 n , p - 1 ) ( mod p 3 ) , 2 2 n p - 3 .
6: 26.7 Set Partitions: Bell Numbers
For S ( n , k ) see §26.8(i). …
26.7.2 B ( n ) = k = 0 n S ( n , k ) ,
7: 26.14 Permutations: Order Notation
In this subsection S ( n , k ) is again the Stirling number of the second kind26.8), and B m is the m th Bernoulli number24.2(i)). …
26.14.7 n k = j = 0 n - k ( - 1 ) n - k - j j ! ( n - j k ) S ( n , j ) ,
26.14.12 S ( n , m ) = 1 m ! k = 0 n - 1 n k ( k n - m ) , n m , n 1 .
8: 26.13 Permutations: Cycle Notation
The Stirling cycle numbers of the first kind, denoted by [ n k ] , count the number of permutations of { 1 , 2 , , n } with exactly k cycles. They are related to Stirling numbers of the first kind by
26.13.3 [ n k ] = | s ( n , k ) | .
9: 24.16 Generalizations
24.16.8 β n ( λ ) = n ! b n λ n + k = 1 n / 2 n 2 k B 2 k s ( n - 1 , 2 k - 1 ) λ n - 2 k , n = 2 , 3 , .
Here s ( n , m ) again denotes the Stirling number of the first kind. …
10: 26.15 Permutations: Matrix Notation