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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: 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 } . …
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: 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
5: 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 …
6: 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. …
7: 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. … 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. …
8: 32.14 Combinatorics
Let S N be the group of permutations 𝝅 of the numbers 1 , 2 , , N 26.2). …
32.14.1 lim N Prob ( N ( 𝝅 ) 2 N N 1 / 6 s ) = F ( s ) ,
9: 26.17 The Twelvefold Way
§26.17 The Twelvefold Way
10: 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. …