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11: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions
M 2 is the number of permutations of { 1 , 2 , , n } with a 1 cycles of length 1, a 2 cycles of length 2, , and a n cycles of length n :
26.4.7 M 2 = n ! 1 a 1 ( a 1 ! )  2 a 2 ( a 2 ! ) n a n ( a n ! ) .
(The empty set is considered to have one permutation consisting of no cycles.) …
12: 34.7 Basic Properties: 9 j Symbol
The 9 j symbol has symmetry properties with respect to permutation of columns, permutation of rows, and transposition of rows and columns; these relate 72 independent 9 j symbols. Even (cyclic) permutations of either columns or rows, as well as transpositions, leave the 9 j symbol unchanged. Odd permutations of columns or rows introduce a phase factor ( 1 ) R , where R is the sum of all arguments of the 9 j symbol. …
13: 19.21 Connection Formulas
Let y , z , and p be positive and distinct, and permute y and z to ensure that y does not lie between z and p . … Because R G is completely symmetric, x , y , z can be permuted on the right-hand side of (19.21.10) so that ( x z ) ( y z ) 0 if the variables are real, thereby avoiding cancellations when R G is calculated from R F and R D (see §19.36(i)). …where both summations extend over the three cyclic permutations of x , y , z . … and x , y , z may be permuted. … For each value of p , permutation of x , y , z produces three values of q , one of which lies in the same region as p and two lie in the other region of the same type. …
14: 22.9 Cyclic Identities
These identities are cyclic in the sense that each of the indices m , n in the first product of, for example, the form s m , p ( 4 ) s n , p ( 4 ) are simultaneously permuted in the cyclic order: m m + 1 m + 2 p 1 2 m 1 ; n n + 1 n + 2 p 1 2 n 1 . … …
15: 19.23 Integral Representations
With l 1 , l 2 , l 3 denoting any permutation of sin θ cos ϕ , sin θ sin ϕ , cos θ , …
16: 19.16 Definitions
which is homogeneous and of degree a in the z ’s, and unchanged when the same permutation is applied to both sets of subscripts 1 , , n . … …
17: 19.26 Addition Theorems
with corresponding equations for y + μ and z + μ obtained by permuting x , y , z . …with x and y obtained by permuting x , y , and z . … The equations inverse to z + λ = ( z + x ) ( z + y ) and the two other equations obtained by permuting x , y , z (see (19.26.19)) are …
18: 19.14 Reduction of General Elliptic Integrals
In (19.14.4) 0 y < x , each quadratic polynomial is positive on the interval ( y , x ) , and α , β , γ is a permutation of 0 , a 1 b 2 , a 2 b 1 (not all 0 by assumption) such that α β γ . …
19: 19.33 Triaxial Ellipsoids
where L a and L b are obtained from L c by permutation of a , b , and c . …
20: 34.2 Definition: 3 j Symbol
where r , s , t is any permutation of 1 , 2 , 3 . …