Introductory Combinatorics

Richard A. Brualdi

Chapter 9 Systems of Distinct Representatives - all with Video Answers

Educators

Chapter Questions

Problem 1

Consider the chessboard $B$ with forbidden positions shown in Figure 9.4. Construct the rook family $\mathcal{A}=\left(A_{1}, A_{2}, A_{3}, A_{4}, A_{5}, A_{6}\right)$ of subsets of $\{1,2,3,4,5,6\}$ of this board. Find six positions for six nonattacking rooks on $B$ and the corresponding SDR of $\mathcal{A}$.

Mark Lozano

Numerade Educator

02:19

Problem 2

Construct the domino family $\mathcal{A}$ of subsets of the black squares associated with the white squares of the board $B$ in Figure 9.4. (Consider the square in the upper left corner to be white.) Determine a tiling of this board and the associated SDR of $\mathcal{A}$.

Aadit Sharma

Numerade Educator

01:13

Problem 3

Give an example of a family $\mathcal{A}$ of sets that is not the domino family of any board.

Nick Johnson

Numerade Educator

02:31

Problem 4

Consider an $m$ -by- $n$ chessboard in which both $m$ and $n$ are odd. The board has one more square of one color, say, black, than of white. Show that, if exactly one black square is forbidden on the board, the resulting board has a tiling with dominoes.

Nick Johnson

Numerade Educator

02:31

Problem 5

Consider an $m$ -by-n chessboard, where at least one of $m$ and $n$ is even. The board has an equal number of white and black squares. Show that if $m$ and $n$ are at least 2 and if exactly one white and exactly one black square are forbidden. the resulting board has a tiling with dominoes.

Nick Johnson

Numerade Educator

View

Problem 6

A corporation has seven available positions $y_{1}, y_{2}, \ldots, y_{7}$ and there are ten applicants $x_{1}, x_{2}, \ldots, x_{10}$. The set of positions each applicant is qualified for is given, respectively, by $\left\{y_{1}, y_{2}, y_{6}\right\},\left\{y_{2}, y_{6}, y_{7}\right\},\left\{y_{3}, y_{4}\right\},\left\{y_{1}, y_{5}\right\},\left\{y_{6}, y_{7}\right\},\left\{y_{3}\right\}$,
$\left\{y_{2}, y_{3}\right\},\left\{y_{1}, y_{3}\right\},\left\{y_{1}\right\},\left\{y_{5}\right\} .$ Determine the largest number of positions that can be filled by the qualified applicants and justify your answer.

Victor Salazar

Numerade Educator

03:14

Problem 7

Let $\mathcal{A}=\left(A_{1}, A_{2}, A_{3}, A_{4}, A_{5}, A_{6}\right)$, where $A_{1}=\{a, b, c\}, A_{2}=\{a, b, c, d, e\}, A_{3}=\{a, b\}$
$A_{4}=\{b, c\}, A_{5}=\{a\}, A_{6}=\{a, c, e\}$
Does the family $\mathcal{A}$ have an SDR? If not, what is the largest number of sets in the family with an SDR?

Nick Johnson

Numerade Educator

00:54

Problem 8

Let $\mathcal{A}=\left(A_{1}, A_{2}, A_{3}, A_{4}, A_{5}, A_{6}\right)$, where
$$\begin{array}{l}A_{1}=\{1,2\}, A_{2}=\{2,3\}, A_{3}=\{3,4\}, \\
A_{4}=\{4,5\}, A_{5}=\{5,6\}, A_{6}=\{6,1\} .\end{array}
$$
Determine the number of different SDRs that $\mathcal{A}$ has. Generalize to $n$ sets.

Jeyasree R T

Numerade Educator

03:14

Problem 9

Let $\mathcal{A}=\left(A_{1}, A_{2}, \ldots, A_{n}\right)$ be a family of sets with an SDR. Let $x$ be an element of $A_{1}$. Prove that there is an SDR containing $x$, but show by example that it may not be possible to find an $S D R$ in which $x$ represents $A_{1}$.

Nick Johnson

Numerade Educator

03:14

Problem 10

Suppose $\mathcal{A}=\left(A_{1}, A_{2}, \ldots, A_{n}\right)$ is a family of sets that "more than satisfies" the marriage condition. More precisely, suppose that
$$\left|A_{i_{1}} \cup A_{i 2} \cup \cdots \cup A_{i_{k}}\right| \geq k+1$$
for each $k=1,2, \ldots, n$ and each choice of $k$ distinct indices $i_{1}, i_{2}, \ldots, i_{k} .$ Let $x$ be an element of $A_{1}$. Prove that $\mathcal{A}$ has an SDR in which $x$ represents $A_{1}$.

Nick Johnson

Numerade Educator

03:30

Problem 11

Let $n>1$, and let $\mathcal{A}=\left(A_{1}, A_{2}, \ldots, A_{n}\right)$ be the family of subsets of $\{1,2, \ldots, n\}$, where
$$A_{i}=\{1,2, \ldots, n\}-\{i\}, \quad(i=1,2, \ldots, n)$$
Prove that $\mathcal{A}$ has an SDR and that the number of SDRs is the $n$ th derangement number $D_{n}$.

Wen Zheng

Numerade Educator

02:31

Problem 12

Consider a board with forbidden positions which has the property that, if a square is forbidden, so is every square to its right in its row and every square below it in its column. Prove that the chessboard has a tiling by dominoes it and only if the number of allowable white squares equals the number of allowable black squares.

Nick Johnson

Numerade Educator

View

Problem 13

Let $A$ be a matrix with $n$ columns, with integer entries taken from the set. $S=\{1,2, \ldots, k\} .$ Assume that each integer $i$ in $S$ occurs exactly $n r_{i}$ times in $A$, where $r_{i}$ is an integer. Prove that it is possible to permute the entries in each row of $A$ to obtain a matrix $B$ in which each integer $i$ in $S$ appears $r_{i}$ times in each column. ${ }^{14}$

Nick Johnson

Numerade Educator

02:21

Problem 14

Let $\mathcal{A}=\left(A_{1}, A_{2}, \ldots, A_{m}\right)$ be a family of subsets of a set $Y=\left\{y_{1}, y_{2}, \ldots, y_{n}\right\} .$
Suppose that there is a positive integer $p$ such that each set of $\mathcal{A}$ contains at least $p$ elements, and each element in $Y$ is contained in at most $p$ sets of $\mathcal{A}$. By counting in two different ways, prove that $n \geq m$.

Nick Johnson

Numerade Educator

05:24

Problem 15

Let $p$ be a positive integer, and let $\mathcal{A}=\left(A_{1}, A_{2}, \ldots, A_{n}\right)$ be a family of $n$ subsets of the set $Y=\left\{y_{1}, y_{2}, \ldots, y_{n}\right\}$ of $n$ elements. Suppose that each set $A_{i}$ of $\mathcal{A}$ contains exactly $p$ elements of $Y$, and each element $y_{j}$ of $Y$ is contained in exactly $p$ sets of $\mathcal{A}$. Prove that $\mathcal{A}$ has an SDR. Reformulate this problem in terms of nonattacking rooks on a board with forbidden positions.

Aymara Gallardo

Numerade Educator

View

Problem 16

Find a 2-by-2 preferential ranking matrix for which both complete marriages are stable.

Nick Johnson

Numerade Educator

View

Problem 17

Consider a preferential ranking matrix in which woman $A$ ranks man $a$ first, and man $a$ ranks $A$ first. Show that, in every stable marriage, $A$ is paired with $a .$

Victor Salazar

Numerade Educator

02:00

Problem 18

Consider the preferential ranking matrix $\left[\begin{array}{ccccc}1, n & 2, n-1 & 3, n-2 & \cdots & n, 1 \\ n, 1 & 1, n & 2, n-1 & \cdots & n-1,2 \\ n-1,2 & n, 1 & 1, n & \cdots & n-2,3 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 3, n-2 & 4, n-3 & 5, n-4 & \cdots & 2, n-1 \\ 2, n-1 & 3, n-2 & 4, n-3 & \cdots & 1, n\end{array}\right]$
Prove that, for each $k=1,2, \ldots, n$, the complete marriage in which each woman gets her $k$ th choice is stable.

Manik Pulyani

Numerade Educator

02:00

Problem 19

Use the deferred acceptance algorithm to obtain both the women-optimal and men-optimal stable complete marriages for the preferential ranking matrix
$A$$B$$C$
$D$$\left[\begin{array}{llll}1,3 & 2,3 & 3,2 & 4,3 \\ 1,4 & 4,1 & 3,3 & 2,2 \\ 2,2 & 1,4 & 3,4 & 4,1 \\ 4,1 & 2,2 & 3,1 & 1,4\end{array}\right]$.
Conclude that, for the given preferential ranking matrix, there is only one stable complete marriage.

Manik Pulyani

Numerade Educator

01:16

Problem 20

Prove that in every application of the deferred acceptance algorithm with $n$ women and $n$ men, there are at most $n^{2}-n+1$ proposals.

Shahab Ullah

Numerade Educator

01:08

Problem 21

Extend the deferred acceptance algorithm to the case in which there are more men than women. In such a case, not all of the men will get partners.

Maxime Rossetti

Numerade Educator

03:55

Problem 22

Show, by using Exercise 19 , that it is possible that in no stable complete marriage does any person get his or her first choice.

Ankit Gupta

Numerade Educator

04:35

Problem 23

Apply the deferred acceptance algorithm to obtain a stable complete marriage for the preferential ranking matrix
$A$$B$$C$
$D$$\left[\begin{array}{llll}1,3 & 2,2 & 3,1 & 4,3 \\ 1,4 & 2,3 & 3,2 & 4,4 \\ 3,1 & 1,4 & 2,3 & 4,2 \\ 2,2 & 3,1 & 1,4 & 4,1\end{array}\right]$

Victor Salazar

Numerade Educator

03:21

Problem 24

Consider an $n$ -by-n board in which there is a nonnegative number $a_{i j}$ in the square in row $i$ and column $j,(1 \leq i, j \leq n)$. Assume that the sum of the numbers in each row and in each column equals 1. Prove that it is possible to place $n$ nonattacking rooks on the board at positions occupied by positive numbers.

Prathan Jarupoonphol

Numerade Educator

01:32

Problem 25

Apply the deferred-acceptance algorithm to obtain a stable marriage for the preferential ranking matrix
$\left[\begin{array}{llllll}1,4 & 2,3 & 3,6 & 4,2 & 5,5 & 6,1 \\ 3,1 & 5,2 & 6,5 & 2,6 & 1,3 & 4,4 \\ 5,5 & 3,6 & 6,1 & 4,4 & 2,2 & 1,3 \\ 6,6 & 5,5 & 4,4 & 3,3 & 2,1 & 1,2 \\ 1,3 & 3,1 & 5,2 & 2,5 & 4,4 & 6,6 \\ 4,2 & 5,4 & 6,3 & 1,1 & 2,6 & 3,4\end{array}\right]$
where the rows correspond to $A, B, C, D, E, F$ and the columns correspond to $a, b, c, d, d, f .$

James Chok

Numerade Educator