Explanation Let T denote the total number of inversions for all n ! permutations of { 1 , 2 , ... , n } . Thus, T is the number of triples ( r , s , a 1 a 2 ... a n ) such that a 1 a 2 ... a n is a permutation of { 1 , 2 , ... , n } and { r , s } is an inversion of a 1 a 2 ... a n . The number of choices is given by, Stage To do No of Choices 1 Select the value of r, s ( n 2 ) 2 Select the location of r,s among a 1 a 2 ... a n ( n 2 ) 3 Choose the remaining n − 2 terms among a 1 a 2

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ISBN: 9780134689616

Author: Brualdi, Richard A.

Publisher: Pearson,

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Question

Chapter 4, Problem 59E

To determine

To prove: the total number of inversions of all n! permutations of 1,2,...,n equals 12n!(n2)=n!n(n+1)4.

Expert Solution & Answer

Chapter 4 Solutions

Introductory Combinatorics

Ch. 4 - Prob. 1E Ch. 4 - Determine the mobile integers in . Ch. 4 - Use the algorithm of Section 4.1 to generate the...Ch. 4 - Prove that in the algorithm of Section 4.1, which...Ch. 4 - Let i1i2 … in be a permutation of {1, 2, …, n}...Ch. 4 - Determine the inversion sequences of the following...Ch. 4 - Construct the permutations of {1, 2, …,8} whose...Ch. 4 - How many permutations of {1, 2, 3, 4, 5, 6}...Ch. 4 - Show that the largest number of inversions of a...Ch. 4 - Bring the permutations 256143 and 436251 to 123456...

Ch. 4 - Let S = {x7, x6,…, x1, x0}. Determine the 8-tuples...Ch. 4 - Let S = {x7, x6,…, x1, x0}. Determine the subsets...Ch. 4 - Generate the 5-tuples of 0s and 1s by using the...Ch. 4 - Prob. 14E Ch. 4 - For each of the following subsets of {x7, x6, …,...Ch. 4 - For each of the subsets (a), (b), (c), and (d) in...Ch. 4 - Which subset of {x7, x6, … , x1, x0} is 150th on...Ch. 4 - Build (the corners and edges of) the 4-cube, and...Ch. 4 - Give an example of a noncyclic Gray code of order...Ch. 4 - Prob. 20E Ch. 4 - Construct the reflected Gray code of order 5...Ch. 4 - Prob. 22E Ch. 4 - Determine the immediate successors of the...Ch. 4 - Prob. 24E Ch. 4 - Prob. 26E Ch. 4 - Prob. 27E Ch. 4 - Prob. 28E Ch. 4 - Determine the 7-subset of {1, 2, … , 15} that...Ch. 4 - Generate the inversion sequences of the...Ch. 4 - Prob. 31E Ch. 4 - Generate the 4-permutations of {1, 2, 3, 4, 5,...Ch. 4 - In which position does the subset 2489 occur in...Ch. 4 - Consider the r-subsets of {1, 2, …, n} in...Ch. 4 - The complement of an r-subset A of {1, 2, … , n}...Ch. 4 - Prob. 36E Ch. 4 - Let R′ and R″ be two partial orders on a set X....Ch. 4 - Let (X1, ≤1) and (X2, ≤2) be partially ordered...Ch. 4 - Let (J, ≤) be the partially ordered set with J =...Ch. 4 - Prob. 40E Ch. 4 - Show that a partial order on a finite set is...Ch. 4 - Describe the cover relation for the partial order...Ch. 4 - Prob. 43E Ch. 4 - Prob. 44E Ch. 4 - Prob. 45E Ch. 4 - Let m be a positive integer and define a relation...Ch. 4 - Consider the partial order ≤ on the set X of...Ch. 4 - Prob. 50E Ch. 4 - Let n be a positive integer, and let Xn be the set...Ch. 4 - Verify that a binary n-tuple an − 1, ⋯ ,a1a0 is in...Ch. 4 - Continuing with Exercise 52, show that can be...Ch. 4 - Let (X, ≤) be a finite partially ordered set. By...Ch. 4 - Prob. 56E Ch. 4 - Prob. 57E Ch. 4 - Prob. 58E Ch. 4 - Prob. 59E

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