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Introductory Combinatorics

Richard A. Brualdi

Chapter 14

Polya Counting - all with Video Answers

Educators

Chapter Questions

00:47

Problem 1

Let
$$
f=\left(\begin{array}{llllll}
1 & 2 & 3 & 4 & 5 & 6 \\
6 & 4 & 2 & 1 & 5 & 3
\end{array}\right) \text { and } g=\left(\begin{array}{llllll}
1 & 2 & 3 & 4 & 5 & 6 \\
3 & 5 & 6 & 2 & 4 & 1
\end{array}\right) \text { . }
$$
Determine
(a) $f \circ g$ and $g \circ f$
(b) $f^{-1}$ and $g^{-1}$
(c) $f^{2}, f^{5}$
(d) $f \circ g \circ f$
(e) $g^{3}$ and $f \circ g^{3} \circ f^{-1}$

Erika Bustos
Erika Bustos
Numerade Educator
01:07

Problem 2

Prove that permutation composition is associative: $(f \circ g) \circ h=f \circ(g \circ h)$.

Carson Merrill
Carson Merrill
Numerade Educator
02:17

Problem 3

Determine the symmetry group and corner-symmetry group of an equilateral triangle.

James Schroeder
James Schroeder
Numerade Educator
05:51

Problem 4

Determine the symmetry group and corner-symmetry group of a triangle that is isoceles but not equilateral.

WM
William Mead
Numerade Educator
05:51

Problem 5

Determine the symmetry group and corner-symmetry group of a triangle that is neither equilateral nor isoceles.

WM
William Mead
Numerade Educator
03:40

Problem 6

Determine the symmetry group of a regular tetrahedron. (Hint: There are 12 symmetries.)

James Schroeder
James Schroeder
Numerade Educator
00:39

Problem 7

Determine the corner-symmetry group of a regular tetrahedron.

Allison Knapp
Allison Knapp
Numerade Educator
03:40

Problem 8

Determine the edge-symmetry group of a regular tetrahedron.

James Schroeder
James Schroeder
Numerade Educator
06:41

Problem 9

Determine the face-symmetry group of a regular tetrehedron.

Ely Crowder
Ely Crowder
Numerade Educator
00:28

Problem 10

Determine the symmetry group and the corner-symmetry group of a rectangle that is not a square.

Erika Bustos
Erika Bustos
Numerade Educator
02:12

Problem 11

Compute the corner-symmetry group of a regular hexagon (the dihedral group $D_{6}$ of order 12).

James Schroeder
James Schroeder
Numerade Educator
02:19

Problem 12

Determine all the permutations in the edge-symmetry group of a square.

AG
Ankit Gupta
Numerade Educator
01:38

Problem 13

Let $f$ and $g$ be the permutations in Exercise 1. Consider the coloring $\mathrm{c}=$ $(R, B, B, R, R, R)$ of $1,2,3,4,5,6$ with the colors $R$ and $B$. Determine the following actions on $\mathrm{c}$ :
(a) $f * \mathbf{c}$
(b) $f^{-1} \cdot \mathrm{c}$
(c) $g * \mathbf{c}$
(d) $(g \circ f) * \mathrm{c}$ and $(f \circ g) * \mathrm{c}$
(e) $\left(g^{2} \circ f\right) * \mathbf{c}$

Ashley Volpe
Ashley Volpe
Numerade Educator
01:06

Problem 14

By examining all possibilities, determine the number of nonequivalent colorings of the corners of an equilateral triangle with the colors red and blue. (Then do so with the colors red, white, and blue.)

Heather Zimmers
Heather Zimmers
Numerade Educator
00:39

Problem 15

By examining all possibilities, determine the number of nonequivalent colorings of the corners of a regular tetrahedron with the colors red and blue. (Then do so with the colors red, white, and blue.)

Allison Knapp
Allison Knapp
Numerade Educator
01:24

Problem 16

Characterize the cycle factorizations of those permutations $f$ in $S_{n}$ for which $f^{-1}=f$, that is, for which $f^{2}=\iota$.

Vishnu P
Vishnu P
Numerade Educator
01:34

Problem 17

In Section $14.2$ it is established that there are eight nonequivalent colorings of the corners of a regular pentagon with the colors red and blue. Explicitly determine eight nonequivalent colorings.

Chris Trentman
Chris Trentman
Numerade Educator
02:06

Problem 18

Use Theorem 14.2.3 to determine the number of nonequivalent colorings of the corners of a square with $p$ colors.

Aman Gupta
Aman Gupta
Numerade Educator
13:31

Problem 19

Use Theorem $14.2 .3$ to determine the number of nonequivalent colorings of the corners of an equilateral triangle with the colors red and blue. Do the same with $p$ colors (cf. Exercise 3).

TA
Tattwamasi Amrutam
Numerade Educator
01:08

Problem 20

Use Theorem 14.2.3 to determine the number of nonequivalent colorings of the corners of a triangle that is isoceles, but not equilateral, with the colors red and blue. Do the same with $p$ colors (cf. Exercise 4).

Chris Trentman
Chris Trentman
Numerade Educator
01:08

Problem 21

Use Theorem $14.2 .3$ to determine the number of nonequivalent colorings of the corners of a triangle that is neither equilateral nor isoceles, with the colors red and blue. Do the same With $p$ colors (cf. Exercise 5).

Chris Trentman
Chris Trentman
Numerade Educator
01:15

Problem 22

Use Theorem 14.2.3 to determine the number of nonequivalent colorings of the corners of a rectangle that is not a square with the colors red and blue. Do the same with $p$ colors (cf. Exercise 10).

Tiffany Tran
Tiffany Tran
Numerade Educator
02:18

Problem 23

A (one-sided) marked domino is a piece consisting of two squares joined along an edge, where each square on one side of the piece is marked with $0,1,2,3,4,5$, or 6 dots. The two squares of a marked domino may receive the same number of dots.
(a) Use Theorem 14.2.3 to determine the number of different marked dominoes.
(b) How many different marked dominoes are there if we are allowed to mark the squares with $0,1, \ldots, p-1$, or $p$ dots?

Charles Carter
Charles Carter
Numerade Educator
02:18

Problem 24

A two-sided marked domino is a piece consisting of two squares joined along an edge, where each square on both sides of the piece is marked with $0,1,2,3$, 4,5, or 6 dots.
(a) Use Theorem 14.2.3 to determine the number of different two sided markeddominoes.
(b) How many different two-sided marked dominoes are there if we are allowed to mark the squares with $0,1, \ldots, p-1$, or $p$ dots?

Charles Carter
Charles Carter
Numerade Educator
01:50

Problem 25

How many different necklaces are there that contain three red and two blue beads?

Gregory Higby
Gregory Higby
Numerade Educator
02:07

Problem 26

How many different necklaces are there that contain four red and three blue beads?

Gregory Higby
Gregory Higby
Numerade Educator
01:37

Problem 27

Determine the cycle factorization of the permutations $f$ and $g$ in Exercise 1 .

Lindsay El
Lindsay El
Numerade Educator
02:00

Problem 28

Let $f$ be a permutation of a set $X$. Give a simple algorithm for finding the cycle factorization of $f^{-1}$ from the cycle factorization of $f$.

Victor Salazar
Victor Salazar
Numerade Educator
02:32

Problem 29

Determine the cycle factorization of each permutation in the dihedral group $D_{6}$ (cf. Exercise 11).

Victor Salazar
Victor Salazar
Numerade Educator
01:18

Problem 30

Determine permutations $f$ and $g$ of the same set $X$ such that $f$ and $g$ each have two cycles in their cycle factorizations but $f \circ g$ has only one.

Aymara Gallardo
Aymara Gallardo
Numerade Educator
02:49

Problem 31

Show that the number of nonequivalent colorings of the corners of a regular 5 -gon with $p$ colors is
$$
\frac{p\left(p^{2}+4\right)\left(p^{2}+1\right)}{10}
$$

Sarah Gift
Sarah Gift
Numerade Educator
00:14

Problem 32

Determine the number of nonequivalent colorings of the corners of a regular hexagon with the colors red, white and blue (cf. Exercise 29).

Amrita Bhasin
Amrita Bhasin
Numerade Educator
07:07

Problem 33

Prove that a permutation and its inverse have the same type (cf. Exercise 28).

WM
William Mead
Numerade Educator
03:06

Problem 34

Let $e_{1}, e_{2}, \ldots, e_{n}$ be nonnegative integers such that $1 e_{1}+2 e_{2}+\cdots+n e_{n}=$
n. Show how to construct a permutation $f$ of the set $\{1,2, \ldots, n\}$ such that $\operatorname{type}(f)=\left(e_{1}, e_{2}, \ldots, e_{n}\right)$

Sriparna Bhattacharjee
Sriparna Bhattacharjee
Numerade Educator
01:36

Problem 35

Determine the number of nonequivalent colorings of the corners of a regular 6 -gon with $k$ colors (cf. Exercise 29).

Christopher Stanley
Christopher Stanley
Numerade Educator
02:19

Problem 36

Determine the number of nonequivalent colorings of the corners of a regular $5-$ gon with the colors red, white, and blue in which two corners are colored red, two are colored white, and one is colored blue.

Christopher Stanley
Christopher Stanley
Numerade Educator
01:36

Problem 37

Determine the number of nonequivalent colorings of the corners of a regular 8 . gon with colors red, white, and blue under the action of the corner symmetry group of the 8 -gon.

Christopher Stanley
Christopher Stanley
Numerade Educator
01:59

Problem 38

A two-sided triomino is a 1 by 3 board of three squares with each square (six in in all because of the two sides) colored with one of the colors red, white, blue, green, and yellow (squares on opposite sides may be colored differently). How many nonequivalent two-sided triominoes are there?

Sheryl Ezze
Sheryl Ezze
Numerade Educator
00:28

Problem 39

A two-sided 4 -omino is a 1 -by-4 board of four squares with each square (eight in in all because of the two sides) colored with one of the colors red, white, blue, green, and yellow (squares on opposite sides may be colored differently). How many nonequivalent two-sided 4-ominoes are there?

Erika Bustos
Erika Bustos
Numerade Educator
00:45

Problem 40

A two-sided $n$ -omino is a 1-by-n board of $n$ squares with each square $(2 n$ in in all because of the two sides) colored with one of $p$ given colors (squares on opposite sides may be colored differently). How many nonequivalent two-sided $n$ -ominoes are there?

AG
Ankit Gupta
Numerade Educator
00:25

Problem 41

Determine the cycle index of the dihedral group $D_{6}$ (cf. Exercise 29).

Ashly Sunny
Ashly Sunny
Numerade Educator
01:38

Problem 42

Determine the generating function for nonequivalent colorings of the corners of a regular hexagon with two colors and also with three colors (cf. Exercise 41).

Christopher Stanley
Christopher Stanley
Numerade Educator
00:54

Problem 43

Determine the cycle index of the edge-symmetry group of a square.

Charles Carter
Charles Carter
Numerade Educator
01:34

Problem 44

Determine the generating function for nonequivalent colorings of the edges of a square with the colors red and blue. How many nonequivalent colorings are there with $k$ colors (cf. Exercise 43)?

Chris Trentman
Chris Trentman
Numerade Educator
03:06

Problem 45

Let $n$ be an odd prime number. Prove that each of the permutations, $\rho_{n}, \rho_{n}^{2}, \ldots, \rho_{n}^{n}$ of $\{1,2, \ldots, n\}$ is an $n$ -cycle. (Recall that $\rho_{n}$ is the permutation that sends 1 to 2, 2 to $3, \ldots, n-1$ to $n$, and $n$ to $1 .$ )

Sriparna Bhattacharjee
Sriparna Bhattacharjee
Numerade Educator
02:22

Problem 46

Let $n$ be a prime number. Determine the number of different necklaces that can be made from $n$ beads of $k$ different colors.

James Chok
James Chok
Numerade Educator
01:38

Problem 47

The nine squares of a 3 -by-3 chessboard are to be colored red and blue. The chessboard is free to rotate but cannot be flipped over. Determine the generating function for the number of nonequivalent colorings and the total number of nonequivalent colorings.

Christopher Stanley
Christopher Stanley
Numerade Educator
01:39

Problem 48

A stained glass window in the form of a 3 -by-3 chessboard has nine squares, each of which is colored red or blue (the colors are transparent and the window can be looked at from either side). Determine the generating function for the number of different stained glass windows and the total number of stained glass windows.

Christopher Stanley
Christopher Stanley
Numerade Educator
00:40

Problem 49

Repeat Exercise 48 for stained glass windows in the form of a 4 -by-4 chessboard with 16 squares.

Prathan Jarupoonphol
Prathan Jarupoonphol
Numerade Educator
01:26

Problem 50

Find the generating function for the different necklaces that can be made with $p$ beads each of color red or blue if $p$ is a prime number (cf. Exercise 46$)$.

Melissa Lupinacci
Melissa Lupinacci
Numerade Educator
01:18

Problem 51

Determine the cycle index of the dihedral group $D_{2 p}$, where $p$ is a prime number.

James Chok
James Chok
Numerade Educator
01:42

Problem 52

Find the generating function for the different necklaces that can be made with $2 p$ beads each of color red or blue if $p$ is a prime number.

Prathan Jarupoonphol
Prathan Jarupoonphol
Numerade Educator
01:00

Problem 53

Ten balls are stacked in a triangular array with 1 atop 2 atop 3 atop 4. (Think of billiards.) The triangular array is free to rotate. Find the generating function for the number of nonequivalent colorings with the colors red and blue. Find the generating function if we are also allowed to turn over the array.

WZ
Wen Zheng
Numerade Educator
00:55

Problem 54

Use Theorem 14.3.3 to determine the generating function for nonisomorphic graphs of order $5 .$ (Hint This exercise will require some work and is a fitting last exercise. We need to obtain the cycle index of the group $S_{5}^{(2)}$ of permutations of the set $X$ of 10 unordered pairs of distinct integers from $\{1,2,3,4,5\}$ (the possible edges of a graph of order 5 ). First, compute the number of permutations $f$ of $S_{5}$ of each type. Then use the fact that the type of $f$ as a permutation of $X$ depends only on the type of $f$ as a permutation of $\{1,2,3,4,5\} .$.)

Clarissa Noh
Clarissa Noh
Numerade Educator