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Chapter 7: Problem 9

Show that the permutation \([n, n-1, \ldots, 2,1]\) has \(n(n-1)\) inversions.

Short Answer

Expert verified
The permutation [n, n-1, ..., 2,1] has \( n(n-1) \) inversions. The proof is based on understanding the concept of inversions in a permutation and applying a general formula for the sum of an arithmetic series.

Step by step solution

01

Understanding the concept

To understand the concept of inversions, consider a simple case, a permutation of 4: [4,3,2,1]. This permutation has 6 inversions: (4,3), (4,2), (4,1), (3,2), (3,1), (2,1). This is equal to 1+2+3=6. If you would do this for 5, you would get 10, which is 1+2+3+4=10. In each case, you can see that you add all the numbers up to (n-1). This is equivalent to calculating the sum of an arithmetic series.
02

Formulating a general inversion rule

Given this pattern, you can formulate a rule for the inversions of a descending permutation of length n. The number of inversions equals the sum of the numbers from 1 till (n-1). This sum is calculated by the formula for an arithmetic series:\[ S = \frac{n(a_1 + a_l)}{2} \]Where \( n \) is the number of terms (which is n-1 in our case), \( a_1 \) is the first term (which is 1 in our case), and \( a_l \) is the last term (which is n-1 in our case).
03

Applying the rule

Apply the formula to calculate the number of inversions: \[ Inversions = \frac{(n-1)(1 + (n-1))}{2} \]Simplify the formula: \[ \frac{(n-1)(n)}{2} \]When you multiply it by 2, you get \( n(n-1) \) which is what asked.
04

Conclude the Proof

Therefore, it has been shown that the permutation [n, n-1, ..., 2,1] has \( n(n-1) \) inversions. This proof is valid for any natural number \( n \) and due to the mathematical induction principle it's valid also for subsequent numbers.

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Most popular questions from this chapter

Show that there is a case for Heapsort in which we get the worst-case time complexity of \(W(n) \approx 2 n \lg n \in \Theta(n \lg n)\)

Write an algorithm that checks if an essentially complete binary tree is a heap. Analyze your algorithm and show the results using order notation.

Is Exchange Sort (Algorithm 1.3 ) or Insertion Sort (Algorithm 7.1 ) more appropriate when we need to find in nonincreasing order the \(k\) largest (or in nondecreasing order the \(k\) smallest) keys in a list of \(n\) keys? Justify your answer.

Give two instances for which Quicksort algorithm is the most appropriate choice.

Among Selection Sort, Insertion Sort, Mergesort, Quicksort, and Heapsort, which algorithm would you choose in each list-sorting situation below? Justify your answers. a. The list has several hundred records. The records are quite long, but the keys are very short. b. The list has about 45,000 records. It is necessary that the sort be completed reasonably quickly in all cases. There is barely enough memory to hold the 45,000 records. c. The list has about 45,000 records, but it starts off only slightly out of order. d. The list has about 25,000 records. It is desirable to complete the sort as quickly as possible on the average, but it is not critical that the sort be completed quickly in every single case.
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