Introductory Combinatorics
Introductory Combinatorics
5th Edition
ISBN: 9780136020400
Author: Richard A. Brualdi
Publisher: Prentice Hall
Question
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Chapter 4, Problem 1E
To determine

The permutation of {1,2,3,4,5} which follows 31524 and comes before 31524.

Expert Solution & Answer
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Answer to Problem 1E

The permutation 31524 is followed by 35124_ and preceded by 31254_.

Explanation of Solution

Algorithm used:

Begin with 1,2,,n.

While there exists a mobile integer, do the following:

(1) Find the largest mobile integer m.

(2) Switch m and the adjacent integer to which its arrow points.

(3) Switch the direction of all the arrows above integers p with p>m.

Calculation:

The permutations of (1,2,3,4) are listed in 24 rows.

According to the algorithm mentioned above, for each row (1,2,3,.........24), insert 5 in the five possible locations of the given permutation, working from right to left for odd numbered rows and left to right for even numbered rows.

The permutation 3124 appears on row 9 which is odd and so, for this row the 5's are inserted from right to left.

Thus, the permutation 31524 is followed by 35124_ and preceded by 31254_.

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Chapter 4 Solutions

Introductory Combinatorics

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. 14ECh. 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. 20ECh. 4 - Construct the reflected Gray code of order 5...Ch. 4 - Prob. 22ECh. 4 - Determine the immediate successors of the...Ch. 4 - Prob. 24ECh. 4 - Prob. 26ECh. 4 - Prob. 27ECh. 4 - Prob. 28ECh. 4 - Determine the 7-subset of {1, 2, … , 15} that...Ch. 4 - Generate the inversion sequences of the...Ch. 4 - Prob. 31ECh. 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. 36ECh. 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. 40ECh. 4 - Show that a partial order on a finite set is...Ch. 4 - Describe the cover relation for the partial order...Ch. 4 - Prob. 43ECh. 4 - Prob. 44ECh. 4 - Prob. 45ECh. 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. 50ECh. 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. 56ECh. 4 - Prob. 57ECh. 4 - Prob. 58ECh. 4 - Prob. 59E
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