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1: 26.13 Permutations: Cycle Notation
§26.13 Permutations: Cycle Notation
The permutationSee §26.8 for generating functions, recurrence relations, identities, and asymptotic approximations. A derangement is a permutation with no fixed points. … Every permutation is a product of transpositions. …
2: 19.15 Advantages of Symmetry
§19.15 Advantages of Symmetry
The function R a ( b 1 , b 2 , , b n ; z 1 , z 2 , , z n ) (Carlson (1963)) reveals the full permutation symmetry that is partially hidden in F D , and leads to symmetric standard integrals that simplify many aspects of theory, applications, and numerical computation. … Symmetry unifies the Landen transformations of §19.8(ii) with the Gauss transformations of §19.8(iii), as indicated following (19.22.22) and (19.36.9). … …
3: 26.2 Basic Definitions
Permutation
A permutation is a one-to-one and onto function from a non-empty set to itself. If the set consists of the integers 1 through n , a permutation σ can be thought of as a rearrangement of these integers where the integer in position j is σ ( j ) . Thus 231 is the permutation σ ( 1 ) = 2 , σ ( 2 ) = 3 , σ ( 3 ) = 1 . … If, for example, a permutation of the integers 1 through 6 is denoted by 256413 , then the cycles are ( 1 , 2 , 5 ) , ( 3 , 6 ) , and ( 4 ) . …
4: Bille C. Carlson
The main theme of Carlson’s mathematical research has been to expose previously hidden permutation symmetries that can eliminate a set of transformations and thereby replace many formulas by a few. …This symmetry led to the development of symmetric elliptic integrals, which are free from the transformations of modulus and amplitude that complicate the Legendre theory. … In Symmetry in c, d, n of Jacobian elliptic functions (2004) he found a previously hidden symmetry in relations between Jacobian elliptic functions, which can now take a form that remains valid when the letters c, d, and n are permuted. …In Permutation symmetry for theta functions (2011) he found an analogous hidden symmetry between theta functions. …
5: 26.14 Permutations: Order Notation
§26.14 Permutations: Order Notation
The permutation 35247816 has two descents: 52 and 81 . … … It is also equal to the number of permutations in 𝔖 n with exactly k + 1 weak excedances. …
§26.14(iii) Identities
6: 20.11 Generalizations and Analogs
§20.11(v) Permutation Symmetry
The importance of these combined theta functions is that sets of twelve equations for the theta functions often can be replaced by corresponding sets of three equations of the combined theta functions, plus permutation symmetry. …
7: 26.16 Multiset Permutations
§26.16 Multiset Permutations
𝔖 S denotes the set of permutations of S for all distinct orderings of the a 1 + a 2 + + a n integers. The number of elements in 𝔖 S is the multinomial coefficient (§26.4) ( a 1 + a 2 + + a n a 1 , a 2 , , a n ) . … The definitions of inversion number and major index can be extended to permutations of a multiset such as 351322453154 𝔖 { 1 2 , 2 2 , 3 3 , 4 2 , 5 3 } . …
8: 34.7 Basic Properties: 9 j Symbol
§34.7(ii) Symmetry
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. For further symmetry properties of the 9 j symbol see Edmonds (1974, pp. 102–103) and Varshalovich et al. (1988, §10.4.1). …
9: 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.) …
Table 26.4.1: Multinomials and partitions.
n m λ M 1 M 2 M 3
5 2 2 1 , 3 1 10 20 10
5 3 1 2 , 3 1 20 20 10
10: 26.15 Permutations: Matrix Notation
§26.15 Permutations: Matrix Notation
The set 𝔖 n 26.13) can be identified with the set of n × n matrices of 0’s and 1’s with exactly one 1 in each row and column. …The permutation 35247816 corresponds to the matrix … The sign of the permutation σ is the sign of the determinant of its matrix representation. … The number of permutations that avoid B is …