Solutions for Introductory Combinatorics
Problem 2E:
Determine the mobile integers in
.
Problem 3E:
Use the algorithm of Section 4.1 to generate the first 50 permutations {1, 2, 3, 4, 5}, starting...Problem 4E:
Prove that in the algorithm of Section 4.1, which generates directly the permutations of {1, 2, …,...Problem 5E:
Let i1i2 … in be a permutation of {1, 2, …, n} with inversion sequence b1, b2, …, b and let k = b1 +...Problem 6E:
Determine the inversion sequences of the following permutations of {1, 2, …, 8}:
35168274
83476215
Problem 7E:
Construct the permutations of {1, 2, …,8} whose inversion sequences are
2, 5, 5, 0, 2, 1, 1, 0
6, 6,...Problem 9E:
Show that the largest number of inversions of a permutation of {1, 2, …, n} equals n(n − 1)/2....Problem 10E:
Bring the permutations 256143 and 436251 to 123456 by successive switches of adjacent numbers.
Problem 11E:
Let S = {x7, x6,…, x1, x0}. Determine the 8-tuples of 0s and 1s corresponding to the following...Problem 12E:
Let S = {x7, x6,…, x1, x0}. Determine the subsets of S corresponding to the following...Problem 13E:
Generate the 5-tuples of 0s and 1s by using the base 2 arithmetic generating scheme and identify...Problem 15E:
For each of the following subsets of {x7, x6, …, x1, x0}, determine the subset that immediately...Problem 16E:
For each of the subsets (a), (b), (c), and (d) in the preceding exercise, determine the subset that...Problem 17E:
Which subset of {x7, x6, … , x1, x0} is 150th on the list of subsets of S when the base 2 arithmetic...Problem 18E:
Build (the corners and edges of) the 4-cube, and indicate the reflected Gray code on it.
Problem 19E:
Give an example of a noncyclic Gray code of order 3.
Problem 21E:
Construct the reflected Gray code of order 5 by
using the inductive definition, and
using the Gray...Problem 23E:
Determine the immediate successors of the following 9-tuples in the reflected Gray code of order...Problem 29E:
Determine the 7-subset of {1, 2, … , 15} that immediately follows 1, 2, 4, 6, 8, 14, 15 in the...Problem 30E:
Generate the inversion sequences of the permutations of {1, 2, 3} in the lexicographic order, and...Problem 32E:
Generate the 4-permutations of {1, 2, 3, 4, 5, 6}.
Problem 33E:
In which position does the subset 2489 occur in the lexicographic order of the 4-subsets of {1, 2,...Problem 34E:
Consider the r-subsets of {1, 2, …, n} in lexicographic order.
What are the first (n − r + 1)...Problem 35E:
The complement of an r-subset A of {1, 2, … , n} is the (n − r)-subset of {1, 2, … , n}, consisting...Problem 37E:
Let R′ and R″ be two partial orders on a set X. Define a new relation R on X by x R y if and only if...Problem 38E:
Let (X1, ≤1) and (X2, ≤2) be partially ordered sets. Define a relation T on the set
by
Prove that...Problem 39E:
Let (J, ≤) be the partially ordered set with J = {0, 1} and with 0 < 1. By identifying the subsets...Problem 41E:
Show that a partial order on a finite set is uniquely determined by its cover relation.
Problem 42E:
Describe the cover relation for the partial order ⊆ on the collection P(X) of all subsets of a set...Problem 46E:
Let m be a positive integer and define a relation R on the set X of all nonnegative integers by a R...Problem 48E:
Consider the partial order ≤ on the set X of positive integers given by "is a divisor of." Let a and...Problem 51E:
Let n be a positive integer, and let Xn be the set of n! permutations of {1, 2, … , n} Let π and σ...Problem 52E:
Verify that a binary n-tuple an − 1, ⋯ ,a1a0 is in place k in the Gray code order list where k is...Problem 53E:
Continuing with Exercise 52, show that can be recovered from by an−1 = bn−1, and for i = 0, 1, … ,...Browse All Chapters of This Textbook
Chapter 1 - What Is Combinatorics?Chapter 2 - Permutations And CombinationsChapter 3 - The Pigeonhole PrincipleChapter 4 - Generating Permutations And CombinationsChapter 5 - The Binomial CoefficientsChapter 6 - The Inclusion-exclusion Principle And ApplicationsChapter 7 - Recurrence Relations And Generating FunctionsChapter 8 - Special Counting SequencesChapter 9 - Systems Of Distinct RepresentativesChapter 10 - Combinatorial Designs
Sample Solutions for this Textbook
We offer sample solutions for Introductory Combinatorics homework problems. See examples below:
To show this, we have to add 3 cases here. Case 1: Assume that one of m and n is even and the second...Procedure used: Multiplication principle: When a task has p outcomes and, no matter what the outcome...Given: The cumulative number of games played on the first n days is denoted by an, where n=1,2,…,77....Algorithm used: Begin with 1←,2←,⋯,n←. While there exists a mobile integer, do the following: (1)...Formula used: The pascal’s triangle formula is: (nk)=n!k!(n−k)!=n(n−1)⋅⋅⋅(n−k+1)k(k−1)⋅⋅⋅1...Suppose the set S={1,2,...,104}. Let A, B, C be the set of integers S that are divisible by 4, 5, 6...Using the mathematical induction and the Fibonacci recurrence. The sequence of numbers...Chapter 8, Problem 1EDefinition used: Let Y be a finite set and A=(A1,A2,…,An) be a family of n subsets of Y. A family...
Definition used: “Let n be a positive integer with n≥2, then Zn={0,1,…,n−1}.” “For any two integers...Definition used: “Two general graphs G=(V,E) and G=(V′,E′) are called isomorphic, provided that...Definition used: Chromatic number: Let G=(V,E) be a graph. A vertex coloring of G is an assignment...The given permutations are, f=(123456642153) and g=(123456356241). Here, (f∘g)(1)=2, (f∘g)(2)=5,...
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