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1: 26.13 Permutations: Cycle Notation
§26.13 Permutations: Cycle Notation
… ►The permutation … ►See §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
… ► denotes the set of permutations of for all distinct orderings of the integers. The number of elements in is the multinomial coefficient (§26.4) . … ►The definitions of inversion number and major index can be extended to permutations of a multiset such as . … ►
26.16.3
3: 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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4: 26.2 Basic Definitions
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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 , a permutation can be thought of as a rearrangement of these integers where the integer in position is . Thus is the permutation , , . … ►If, for example, a permutation of the integers 1 through 6 is denoted by , then the cycles are , , and . …5: 26.14 Permutations: Order Notation
§26.14 Permutations: Order Notation
… ►The permutation has two descents: and . … … ►It is also equal to the number of permutations in with exactly weak excedances. … ►§26.14(iii) Identities
…6: 26.15 Permutations: Matrix Notation
§26.15 Permutations: Matrix Notation
►The set (§26.13) can be identified with the set of matrices of 0’s and 1’s with exactly one 1 in each row and column. …The permutation corresponds to the matrix … ►The sign of the permutation is the sign of the determinant of its matrix representation. … ►The number of permutations that avoid is …7: 14.14 Continued Fractions
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►provided that and do not vanish simultaneously for any .
…again provided and do not vanish simultaneously for any .
8: 19.15 Advantages of Symmetry
§19.15 Advantages of Symmetry
… ►The function (Carlson (1963)) reveals the full permutation symmetry that is partially hidden in , and leads to symmetric standard integrals that simplify many aspects of theory, applications, and numerical computation. …9: Bille C. Carlson
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►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.
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►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.
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