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partitional shifted factorial

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1: 35.4 Partitions and Zonal Polynomials
Also, | κ | denotes k 1 + + k m , the weight of κ ; ( κ ) denotes the number of nonzero k j ; a + κ denotes the vector ( a + k 1 , , a + k m ) . The partitional shifted factorial is given by
35.4.1 [ a ] κ = Γ m ( a + κ ) Γ m ( a ) = j = 1 m ( a - 1 2 ( j - 1 ) ) k j ,
35.4.2 Z κ ( I ) = | κ | !  2 2 | κ | [ m / 2 ] κ 1 j < l ( κ ) ( 2 k j - 2 k l - j + l ) j = 1 ( κ ) ( 2 k j + ( κ ) - j ) !
2: 35.1 Special Notation
a , b

complex variables.

[ a ] κ

partitional shifted factorial35.4(i)).

3: 35.5 Bessel Functions of Matrix Argument
4: 35.6 Confluent Hypergeometric Functions of Matrix Argument
5: 35.8 Generalized Hypergeometric Functions of Matrix Argument
35.8.1 F q p ( a 1 , , a p b 1 , , b q ; T ) = k = 0 1 k ! | κ | = k [ a 1 ] κ [ a p ] κ [ b 1 ] κ [ b q ] κ Z κ ( T ) .
6: 35.7 Gaussian Hypergeometric Function of Matrix Argument
7: 26.12 Plane Partitions
26.12.17 h = 0 r - 1 ( 3 h + 1 ) ! ( r + h ) ! .
8: 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 )
unlabeled unlabeled p k ( n ) { 1 n k 0 n > k p k ( n ) - p k - 1 ( n )
9: 27.14 Unrestricted Partitions
§27.14 Unrestricted Partitions
§27.14(i) Partition Functions
§27.14(iii) Asymptotic Formulas
For example, p ( 10 ) = 42 , p ( 100 ) = 1905 69292 , and p ( 200 ) = 397 29990 29388 . …
§27.14(v) Divisibility Properties
10: 26.8 Set Partitions: Stirling Numbers
§26.8 Set Partitions: Stirling Numbers
S ( n , k ) denotes the Stirling number of the second kind: the number of partitions of { 1 , 2 , , n } into exactly k nonempty subsets. …
26.8.7 k = 0 n s ( n , k ) x k = ( x - n + 1 ) n ,
26.8.10 k = 1 n S ( n , k ) ( x - k + 1 ) k = x n ,