About the Project

multinomial coefficients

AdvancedHelp

(0.000 seconds)

4 matching pages

1: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions
§26.4 Lattice Paths: Multinomial Coefficients and Set Partitions
§26.4(i) Definitions
Table 26.4.1: Multinomials and partitions.
n m λ M 1 M 2 M 3
§26.4(ii) Generating Function
§26.4(iii) Recurrence Relation
2: 26.16 Multiset Permutations
The number of elements in 𝔖 S is the multinomial coefficient26.4) ( a 1 + a 2 + + a n a 1 , a 2 , , a n ) . … Thus inv ( 351322453154 ) = 4 + 8 + 0 + 3 + 1 + 1 + 2 + 3 + 1 + 0 + 1 = 24 , and maj ( 351322453154 ) = 2 + 4 + 8 + 9 + 11 = 34 . The q -multinomial coefficient is defined in terms of Gaussian polynomials (§26.9(ii)) by
26.16.1 [ a 1 + a 2 + + a n a 1 , a 2 , , a n ] q = k = 1 n 1 [ a k + a k + 1 + + a n a k ] q ,
3: 26.1 Special Notation
( m n ) binomial coefficient.
( m n 1 , n 2 , , n k ) multinomial coefficient.
4: 36.8 Convergent Series Expansions
For multinomial power series for Ψ K ( 𝐱 ) , see Connor and Curtis (1982).
36.8.3 3 2 / 3 4 π 2 Ψ ( H ) ( 3 1 / 3 𝐱 ) = Ai ( x ) Ai ( y ) n = 0 ( 3 1 / 3 i z ) n c n ( x ) c n ( y ) n ! + Ai ( x ) Ai ( y ) n = 2 ( 3 1 / 3 i z ) n c n ( x ) d n ( y ) n ! + Ai ( x ) Ai ( y ) n = 2 ( 3 1 / 3 i z ) n d n ( x ) c n ( y ) n ! + Ai ( x ) Ai ( y ) n = 1 ( 3 1 / 3 i z ) n d n ( x ) d n ( y ) n ! ,
36.8.5 f n ( ζ , ζ ¯ ) = c n ( ζ ) c n ( ζ ¯ ) Ai ( ζ ) Bi ( ζ ¯ ) + c n ( ζ ) d n ( ζ ¯ ) Ai ( ζ ) Bi ( ζ ¯ ) + d n ( ζ ) c n ( ζ ¯ ) Ai ( ζ ) Bi ( ζ ¯ ) + d n ( ζ ) d n ( ζ ¯ ) Ai ( ζ ) Bi ( ζ ¯ ) ,