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Exploring Permutations: Cycles, Notation, and Characteristics

Learn about permutations, cyclic permutations, and polycyclic permutations. Understand the dynamics of fixed points and involutions in permutation groups. Solve homework questions related to permutation orders.

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Exploring Permutations: Cycles, Notation, and Characteristics

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  1. Permutations Introduction – Lecture 2

  2. Permutations • As we know a permutation p is a bijective bijektivna mapping of a set A onto itself: p: A  A. Permutations may be multiplied and form the symmetric group Sym(A) = SA, that has n! elements, where n = |A|.

  3. Permutation as a product of disjoint cycles • A permutation can be written as a product of disjoint cycles in a uniqe way.

  4. Example 1 • Example: • p(1) = 2, p(2) = 6, • p(3) = 5, p(4) = 4, • p(5) = 3, p(6) = 1. • Permutation p can be written as: • p = [2,6,5,4,3,1] • p = (1 2 6)(3 5)(4) 2 S6 6 2 3 5 4

  5. Positional Notation p = [2,6,5,4,3,1] Each row of the configuration table of Pappus configuration is a permutation.

  6. Example • Each row of the configuration table of Pappus configuration can be viwed as permutation (written in positional notation.) • a1 = (1)(2)(3)(4)(5)(6)(7)(8)(9) • a2 = (184)(273)(569) • a3=(16385749)

  7. Cyclic Permutation • A permutation composed of a single cycle is called cyclic permutation. • For example, a3 is cyclic.

  8. Polycyclic Permutation • A permutation whose cycles are of equal length is called semi-regular or polycyclic. • Hence a1, a2 and a3 are polycyclic.

  9. Identity Permutation • A polycyclic pemrutation with cycles of length 1 is called identity permutation.

  10. Fix(p) • Let Fix(p) = {x 2 A| p(x) = x} denote the set of fdixed points of permutation p. • fix(p) = |Fix(p)|. • Fixed-point free premutation is called a derangment.

  11. Order of p.

  12. Involutions • Permutation of order 2 is called an involution. • Later we will be interseted in fixed-point free involutions.

  13. Homework • H1. Determine the number of polycylic configurations of degree n and order k. • H2. Show that an ivolution is fixed-point free if and only if it is a polycyclic permutation composed of cycles of length 2. • H3. Determine the number of involutions in Sn.

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