Problem 20.2/20.2. Let f and g denote the permutations, 2 3 4 5 6 7 15...

Transcribed Image Text:

Problem 20.2/20.2. Let f and g denote the permutations, 2 3 4 5 6 7 15 7 2 6 4 PPS PROBLEM SET 9, DUE 30TH MARCH 1 3 (a) 2n = 10? (b) 2n = 12? f= Write down the disjoint cycle notation for the permutations f. 9. 9², 9³, fg, (fg)-¹, and g-¹f-¹. Problem 20.4/20.4*. A pack of 2n cards is shuffled by the interlacing method, which sends cards 1.... 2n. respectively, to the cards 1, n + 1,2, n +2....n, 2n. After how many shuffles will the cards return to their original position, when 67). (c) 2n = 14? (d) 2n = 16? (e) 2n = 24? (f) 2n = 52 (a real pack of cards)? 9= 1 2 3 4 5 6 7 3 1 7 6 4 5 2, Remark 1. (+) Can you make any conjectures for a general n? (this is not for marks, because it's hard to say much in general with your current tools). Problem 20.6/20.6*. Consider the 3 x 3 version of the "fifteen" puzzle. This has a 3 x 3 grid of numbers, one of which is a *. We think of this as a blank square, and we can move any adjacent square into the blank. Starting from the initial configuration, 3 2 1 4 5 6 78 * 1 2 3 4 5 6 7 8 * determine which of the following configurations can be reached: 1 * 2 1 72 3 4 5 64 5. 678 38 * Problem 20.8/20.8. How many onto functions are there from a set of size n +3 to a set of size n? Hint: You'll want to read the problem and its solution to 20.7, to get you thinking in the right way. Problem 16.6/XX (+). (a) Find the number of arrangements of the set {1,....n} where the numbers 1 and 2 appear as neighbors. (b) Let n ≥ 5. Find the number of arrangements of the set {1,...,n} where 1,2,3 appear as neighbors, and in order, and so do the elements 4, 5. Problem 20.2/20.2. Let f and g denote the permutations, 2 3 4 5 6 7 15 7 2 6 4 PPS PROBLEM SET 9, DUE 30TH MARCH 1 3 (a) 2n = 10? (b) 2n = 12? f= Write down the disjoint cycle notation for the permutations f. 9. 9², 9³, fg, (fg)-¹, and g-¹f-¹. Problem 20.4/20.4*. A pack of 2n cards is shuffled by the interlacing method, which sends cards 1.... 2n. respectively, to the cards 1, n + 1,2, n +2....n, 2n. After how many shuffles will the cards return to their original position, when 67). (c) 2n = 14? (d) 2n = 16? (e) 2n = 24? (f) 2n = 52 (a real pack of cards)? 9= 1 2 3 4 5 6 7 3 1 7 6 4 5 2, Remark 1. (+) Can you make any conjectures for a general n? (this is not for marks, because it's hard to say much in general with your current tools). Problem 20.6/20.6*. Consider the 3 x 3 version of the "fifteen" puzzle. This has a 3 x 3 grid of numbers, one of which is a *. We think of this as a blank square, and we can move any adjacent square into the blank. Starting from the initial configuration, 3 2 1 4 5 6 78 * 1 2 3 4 5 6 7 8 * determine which of the following configurations can be reached: 1 * 2 1 72 3 4 5 64 5. 678 38 * Problem 20.8/20.8. How many onto functions are there from a set of size n +3 to a set of size n? Hint: You'll want to read the problem and its solution to 20.7, to get you thinking in the right way. Problem 16.6/XX (+). (a) Find the number of arrangements of the set {1,....n} where the numbers 1 and 2 appear as neighbors. (b) Let n ≥ 5. Find the number of arrangements of the set {1,...,n} where 1,2,3 appear as neighbors, and in order, and so do the elements 4, 5.

Answer rating: 100% (QA)

Problem 202 S The disjoint cycle structure of the permutations f 1 35... View the full answer