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