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Chapter 7: Problem 9

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

Short Answer

Expert verified
The permutation \([n, n-1, \ldots, 2,1]\) has \(\frac{n(n-1)}{2}\) inversions.

Step by step solution

01

Analyzing the permutation

The given permutation is \([n, n-1, \ldots, 2,1]\). This permutation is in decreasing order. This means that for each number in the permutation, all the numbers after it are smaller. So, each number forms an inversion with every number that comes after it in the sequence.
02

Counting inversions for each number

Considering each number in the permutation, the number n has \(n-1\) numbers after it, hence \(n-1\) inversions. The number \(n-1\) has \(n-2\) numbers after it, hence \(n-2\) inversions. This pattern continues until the number 2, which has 1 number after it, hence 1 inversion. The number 1 has no numbers after it, hence it forms 0 inversions.
03

Total inversions

The total inversions in the permutation is therefore the sum of all the inversions caused by each number. This sum is given by \((n-1) + (n-2) + \ldots + 1\), which is a sum of an arithmetic series. The formula to find the sum of an arithmetic series is \(\frac{(firstTerm + lastTerm) \cdot numberOfTerms}{2}\). The first term in our case is \(n-1\), the last term is 1, and the number of terms is \(n-1\). Plugging these values into the formula, we get \(\frac{(n-1 + 1) \cdot (n-1)}{2}\), which simplifies to \(\frac{n(n-1)}{2}\).
04

Conclusion

So, the number of inversions in a permutation \([n, n-1, \ldots, 2,1]\) is indeed \(\frac{n(n-1)}{2}\). This completes our proof.

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Most popular questions from this chapter

Write a linear-time sorting algorithm that sorts a permutation of integers 1 through \(n\), inclusive.

Suppose we have a very large list stored in external memory that needs to be sorted. Assuming that this list is too large for internal memory, what major factor(s) should be considered in designing an external sorting algorithm?

Implement the Insertion Sort algorithm (Algorithm 7.1), run it on your system, and study its best-case, average-case, and worst-case time complexities using several problem instances.

Modify Heapsort so that it stops after it finds the \(k\) largest keys in nonincreasing order. Analyze your algorithm, and show the results using order notation.

Another way to sort a list by exchanging out-of-order keys is called Bubble Sort. Bubble Sort scans adjacent pairs of records and exchanges those found to have out-of-order keys, After the first time through the list, the record with the largest key (or the smallest key) is moved to its proper position. This process is done repeatedly on the remaining, unsorted part of the list until the list is completely sorted. Write the Bubble Sort algorithm. Analyze your algorithm, and show the results using order notation. Compare the performance of the Bubble Sort algorithm against those of Insertion Sort, Exchange Sort, and Selection Sort.
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