11/12/2014 Ma/CS 6a Class 17: More Permutations π¦π π π π¦π π β¦ π₯1 β¦ π₯1 π¦1 π₯2 π¦1 π₯2 π π₯π π π π₯π By Adam Sheffer Reminder: The Permutation Set ππ ο ππ β The set of permutations of βπ = 1,2,3, β¦ , π . ο The set π3 : 1 2 3 β β β 1 2 3 1 2 3 β β β 1 3 2 1 2 β β 2 1 3 β 3 1 2 3 β β β 2 3 1 1 2 3 β β β 3 1 2 1 2 β β 3 2 3 β 1 1 11/12/2014 Reminder: Cycle Notation ο We can consider a permutation as a set of cycles. 1 2 3 4 5 6 β β β β β β 5 6 3 2 1 4 1β5 ο 2β6β4 3 We write this permutation as 15 264 3 . Reminder: Classification of Permutations ο Both (1 2 4)(3 5) and (1 2 3)(4 5) are of the same type: one cycle of length 3 and one of length 2. β¦ We denote this type as 2 3 ο In general, we write a type as 1πΌ1 2πΌ2 3πΌ3 4πΌ4 β¦ . 2 11/12/2014 The 15 Puzzle and Permutations ο How a configuration of the puzzle can be described as a permutation? β¦ Denote the missing tile as 16. β¦ The board below corresponds to the permutation 1 16 3 4 6 2 11 10 5 8 7 9 14 12 15 13 The 15 Puzzle Revisited ο What kind of permutations describe a move in the 15 Puzzle? β¦ Permutations that switch 16 with an element that was adjacent to it. 3 11/12/2014 Transpositions ο Transposition: a permutation that interchanges two elements and leaves the rest unchanged. β¦ 1 3 5 24 12345 14325 Decomposing a Cycle ο Problem. Write the cycle (1 2 3) as a composition of transpositions. 123 (1 2)(3) 213 (1 3)(2) 231 123 = 13 12 4 11/12/2014 Decomposing a Cycle (2) ο Problem. Write the cycle (π₯1 π₯2 β¦ π₯π ) as a composition of transpositions. π₯1 π₯2 π₯3 β¦ π₯π (π₯1 π₯2 ) π₯2 π₯1 π₯3 β¦ π₯π (π₯1 π₯3 ) π₯1 π₯2 β¦ π₯π = π₯1 π₯π π₯1 π₯πβ1 β― π₯1 π₯2 π₯2 π₯3 π₯1 β¦ π₯π π₯2 π₯3 π₯4 β¦ π₯1 Decomposing a Permutation ο Problem. Can any permutation be written as a composition of transpositions? ο Yes! Write the permutation in its cycle notation and decompose each cycle. 136 2457 = 16 13 27 25 24 5 11/12/2014 Unique Representation? ο ο Problem. Does every permutation have a unique decomposition into transpositions (up to their order)? 123 456 : β¦ 13 12 46 45 . β¦ 14 16 15 34 24 14 . β¦ Noβ¦ But the decompositions of a permutation have a common property. Composing a Permutation with a Transposition ο Problem. β¦ πΌ β a permutation of ππ that consists of π cycles in its cycle notation. β¦ π β a transposition of ππ . β¦ What can we say about the number of cycles in ππΌ? And of πΌπ? 6 11/12/2014 Solution: Case 1 Write π = π π . ο First, assume that π, π are in the same cycle of πΌ. ο β¦ Write the cycle as π π₯1 π₯2 β¦ π₯π π π¦1 π¦2 β¦ π¦π . β¦ Then ππΌ contains the cycles π π₯1 π₯2 β¦ π₯π and π π¦1 π¦2 β¦ π¦π (and similarly for πΌπ). π β π₯1 π₯1 β π₯2 β¦ π₯π β β β¦ π π β π¦1 π¦1 β¦ π¦π β β β π¦2 β¦ π Solution: Case 1 Write π = π π . ο First, assume that π, π are in the same cycle of πΌ. ο β¦ Write the cycle as π π₯1 π₯2 β¦ π₯π π π¦1 π¦2 β¦ π¦π . β¦ Then ππΌ contains the cycles π π₯1 π₯2 β¦ π₯π and π π¦1 π¦2 β¦ π¦π (and similarly for πΌπ). π¦π π π¦π π π β¦ π₯1 β¦ π₯1 π¦1 π₯2 π¦1 π₯2 π π₯π π π π₯π 7 11/12/2014 Solution: Case 2 Write π = π π . ο Assume that π, π are in different cycles of πΌ. ο β¦ Write the cycles as π π₯1 π₯2 β¦ π₯π and π π¦1 π¦2 β¦ π¦π . β¦ Then ππΌ contains the cycle π π₯1 π₯2 β¦ π₯π π π¦1 π¦2 β¦ π¦π . π β π₯1 π₯1 β π₯2 β¦ π₯π β β β¦ π π β π¦1 π¦1 β¦ π¦π β β β π¦2 β¦ π Solution πΌ β a permutation of ππ that consists of π cycles in its cycle notation. ο π β a transposition of ππ . ο The number of cycles in ππΌ (or πΌπ) is either π + 1 or π β 1. ο 8 11/12/2014 Parity of a Permutation ο Theorem. Consider a permutation πΌ β ππ . Then β¦ Either every decomposition of πΌ into transpos. consists of an even number of elements, β¦ or every such decomposition consists of an odd number of elements. ο 123 456 : β¦ 13 12 46 45 . β¦ 14 16 15 34 24 14 . Proof π β the number of cycles in πΌ. ο (WLOG) Assume that π is even. ο Consider a decomposition πΌ = π1 π2 β― ππ . ο β¦ β¦ β¦ β¦ β¦ ο The number of cycles in ππ is π β 1. The number of cycles in ππβ1 ππ is even. The number of cycles in ππβ2 ππβ1 ππ is odd. β¦ The number of cycles in π1 π2 β― ππ is π. Thus, π has the same parity as π. 9 11/12/2014 Even and Odd Permutations ο We say that a permutation is even or odd according to the parity of the number of transpositions in its decompositions. Parity of Inverse ο Problem. Prove that any permutation πΌ β ππ has the same parity as its inverse πΌ β1 . ο Proof. β¦ Decompose πΌ into transpositions π1 π2 β― ππ . β¦ We have πΌ β1 = ππ β― π2 π1 , since the product of these two permutation is obviously id. 10 11/12/2014 The 15 Puzzle ο Problem. Start with the configuration on the left and move the tiles to obtain the configuration on the right. Solution (Finally!) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Even permutation 1 2 3 4 5 6 7 8 9 10 11 12 13 15 14 16 Odd permutation ο The number of moves is even since β¦ For every time that we move the empty tile left/up, we must move it back right/down. 11 11/12/2014 Solution (Finally!) Even permutation Odd permutation ο ο The number of moves/ transpositions is even. To move from an even transposition to an odd one, there must be an odd number of transpositions. Even and Odd Permutations of π5 Type Example 15 13 2 12 3 122 14 23 5 Number id 1 1 2 (3)(4)(5) (1 2 3)(4)(5) 1 2 (3 4)(5) 1234 5 123 45 12345 10 20 15 30 20 24 Even: 60 Even Odd Even Even Odd Odd Even Odd: 60 12 11/12/2014 Even and Odd Permutations of ππ ο Theorem. For any integer π β₯ 2, half of the permutations of ππ are even and half are odd. Proof π β an arbitrary transposition of ππ . ο If πΌ β ππ is even, then ππΌ is odd. ο If πΌ β ππ is odd, then ππΌ is even. ο For any πΌ β ππ , we have πππΌ = πΌ. ο π defines a bijection between the set of even permutations of ππ and the odd permutations of ππ . ο β¦ Thus, the two sets are of the same size. 13 11/12/2014 Example: The Bijection in π3 ο Let π = 1 2 β π3 . Even 1 2 3 123 321 Odd 12 3 1 23 13 2 12 13 = 123 The End: The First Math Theorem Proved in a TV Script? ο ο ο The 10th episode of 6th season of the TV show Futurama is about people switching bodies. This is a permutation of people, and a property of the permutations is used as a plot twist! You can also see a complete mathematical proof for a second. 14
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