Introductory Combinatorics

Richard A. Brualdi

Chapter 8 Special Counting Sequences - all with Video Answers

Educators

Chapter Questions

Problem 1

Let $2 n$ (equally spaced) points on a circle be chosen. Show that the number of ways to join these points in pairs, so that the resulting $n$ line segments do not. intersect, equals the $n$ th Catalan number $C_{n}$.

Debasish Das

Numerade Educator

03:00

Problem 2

Prove that the number of 2 -by-n arrays
$$\left[\begin{array}{llll}x_{11} & x_{12} & \cdots & x_{1 n} \\
x_{21} & x_{22} & \cdots & x_{2 n}\end{array}\right]$$
that can be made from the numbers $1,2 \ldots, 2 n$ such that
$$\begin{array}{c}
x_{11}<x_{12}<\cdots<x_{1 n} \\x_{21}<x_{22}<\cdots<x_{2 n} \\
x_{11}<x_{21}, x_{12}<x_{22}, \ldots, x_{1 n}<x_{2 n}\end{array}
$$ equals the nth Catalan number, $C_{n}$.

Mitchell Riley

Numerade Educator

17:05

Problem 3

Write out all of the multiplication schemes for four numbers and the triangularization of a convex polygonal region of five sides corresponding to them.

Anthony Ramos

Numerade Educator

02:22

Problem 4

Determine the triangularization of a convex polygonal region corresponding to the following multiplication schemes:
(a) $\left(a_{1} \times\left(\left(\left(a_{2} \times a_{3}\right) \times\left(a_{4} \times a_{5}\right)\right) \times a_{6}\right)\right)$
(b) $\left(\left(\left(a_{1} \times a_{2}\right) \times\left(a_{3} \times\left(a_{4} \times a_{5}\right)\right)\right) \times\left(\left(a_{6} \times a_{7}\right) \times a_{8}\right)\right)$

Akash M

Numerade Educator

22:22

Problem 5

Let $m$ and $n$ be nonnegative integers with $n \geq m .$ There are $m+n$ people in line to get into a theater for which admission is 50 cents. Of the $m+n$ people, $n$ have a $50-$ cent piece and $m$ have a $$\$ 1$$ dollar bill. The box office opens with an empty cash register. Show that the number of ways the people can line up so that change is available when needed is
$$\frac{n-m+1}{n+1}\left(\begin{array}{c}m+n \\
m\end{array}\right)$$

Chris Trentman

Numerade Educator

01:21

Problem 6

Let the sequence $h_{0}, h_{1}, \ldots, h_{n}, \ldots$ be defined by $h_{n}=2 n^{2}-n+3,(n \geq 0)$. Determine the difference table, and find a formula for $\sum_{k=0}^{n} h_{k}$.

Aman Gupta

Numerade Educator

01:03

Problem 7

The general term $h_{n}$ of a sequence is a polynomial in $n$ of degree $3 .$ If the first four entries of the Oth row of its difference table are $1,-1,3,10$, determine $h_{n}$ and a formula for $\sum_{k=0}^{n} h_{k}$.

Trinity Steen

Numerade Educator

01:14

Problem 8

Find the sum of the fifth powers of the first $n$ positive integers.

Tanishq Gupta

Numerade Educator

00:48

Problem 9

Prove that the following formula holds for the $k$ th-order differences of a sequence $h_{0}, h_{1}, \ldots, h_{n}, \ldots:$
$$\Delta^{k} h_{n}=\sum_{j=0}^{k}(-1)^{k-j}\left(\begin{array}{l}k \\j\end{array}\right) h_{n+j}$$

Linh Vu

Numerade Educator

02:38

Problem 10

If $h_{n}$ is a polynomial in $n$ of degree $m$, prove that the constants $c_{0}, c_{1}, \ldots, c_{m}$ such that
$$h_{n}=c_{0}\left(\begin{array}{l}n \\
0\end{array}\right)+c_{1}\left(\begin{array}{l}n \\1
\end{array}\right)+\cdots+c_{m}\left(\begin{array}{c}n \\m\end{array}\right)$$
are uniquely determined. (Cf. Theorem 8.2.2.)

Vikash Ranjan

Numerade Educator

01:53

Problem 11

Compute the Stirling numbers of the second kind $S(8, k),(k=0,1, \ldots, 8)$.

Amy Jiang

Numerade Educator

03:00

Problem 12

Prove that the Stirling numbers of the second kind satisfy the following relations:
(a) $S(n, 1)=1, \quad(n \geq 1)$
(b) $S(n, 2)=2^{n-1}-1, \quad(n \geq 2)$
(c) $S(n, n-1)=\left(\begin{array}{c}n \\ 2\end{array}\right), \quad(n \geq 1)$
(d) $S(n, n-2)=\left(\begin{array}{l}n \\ 3\end{array}\right)+3\left(\begin{array}{l}n \\ 4\end{array}\right) \quad(n \geq 2)$

Mitchell Riley

Numerade Educator

03:13

Problem 13

Let $X$ be a $p$ -element set and let $Y$ be a $k$ -element set. Prove that the number of functions $f: X \rightarrow Y$ which map $X$ onto $Y$ equals
$$k ! S(p, k)=S^{\#}(p, k) .$$

Gideon Idumah

Numerade Educator

01:47

Problem 14

Find and verify a general formula for
$$\sum_{k=0}^{n} k^{p}$$
involving Stirling numbers of the second kind.

Nick Johnson

Numerade Educator

01:06

Problem 15

The number of partitions of a set of $n$ elements into $k$ distinguishable boxes (some of which may be empty) is $k^{n}$. By counting in a different way, prove that
$$k^{n}=\left(\begin{array}{l}
k \\1\end{array}\right) 1 ! S(n, 1)+\left(\begin{array}{l}
k \\2\end{array}\right) 2 ! S(n, 2)+\cdots+\left(\begin{array}{l}k \\
n\end{array}\right) n ! S(n, n)$$
(If $k>n$, define $S(n, k)$ to be $0 .$ )

Wendi Zhao

Numerade Educator

00:31

Problem 16

Compute the Bell number $B_{8}$. (Cf. Exercise 11.)

Amy Jiang

Numerade Educator

10:27

Problem 17

Compute the triangle of Stirling numbers of the first kind $s(n, k)$ up to $n=7$.

Bryan Lynn

Numerade Educator

01:22

Problem 18

Write $[n]_{k}$ as a polynomial in $n$ for $k=5,6$, and 7 .

Sherrie Fenner

Numerade Educator

00:58

Problem 19

Prove that the Stirling numbers of the first kind satisfy the following formulas:
(a) $s(n, 1)=(n-1) !, \quad(n \geq 1)$
(b) $s(n, n-1)=\left(\begin{array}{l}n \\ 2\end{array}\right), \quad(n \geq 1)$

Sriram Soundarrajan

Numerade Educator

02:13

Problem 20

Verify that $[n]_{n}=n !$, and write $n !$ as a polynomial in $n$ using the Stirling numbers of the first kind. Do this explicitly for $n=6$.

Ankit Gupta

Numerade Educator

03:56

Problem 21

For each integer $n=1,2,3,4,5$, construct the diagram of the set $\mathcal{P}_{n}$ of partitions of $n$, partially ordered by majorization.

Akash Goyal

Numerade Educator

00:26

Problem 22

. (a) Calculate the partition number $p_{6}$ and construct the diagram of the set $\mathcal{P}_{6}$, partially ordered by majorization.
(b) Calculate the partition number $p_{7}$ and construct the diagram of the set $\mathcal{P}_{7}$, partially ordered by majorization.

Ankit Gupta

Numerade Educator

02:39

Problem 23

A total order on a finite set has a unique maximal element (a largest element) and a unique minimal element (a smallest element). What are the largest partition and smallest partition in the lexicographic order on $\mathcal{P}_{n}$ (a total order)?

Bryan Lynn

Numerade Educator

07:04

Problem 24

A partial order on a finite set may have many maximal elements and minimal elements, In the set $\mathcal{P}_{n}$ of partitions of $n$ partially ordered by majorization, prove that there is a unique maximal element and a unique minimal element.

Joel Mueller

Numerade Educator

01:06

Problem 25

Let $t_{1}, t_{2}, \ldots, t_{m}$ be distinct positive integers, and let
$$q_{n}=q_{n}\left(t_{1}, t_{2}, \ldots, t_{m}\right)$$
equal the number of partitions of $n$ in which all parts are taken from $t_{1}, t_{2}, \ldots, t_{m}$. Define $q_{0}=1$. Show that the generating function for $q_{0}, q_{1}, \ldots, q_{n}, \ldots$ is$$\prod_{k=1}^{m}\left(1-x^{t_{k}}\right)^{-1} .$$

Wendi Zhao

Numerade Educator

01:51

Problem 26

For each integer $n>2$, determine a self-conjugate partition of $n$ that has at least two parts.

Mohamed Mohamed

Numerade Educator

01:51

Problem 27

For each integer $n>2$, determine a self-conjugate partition of $n$ that has at least two parts.

Mohamed Mohamed

Numerade Educator

02:14

Problem 28

Prove that conjugation reverses the order of majorization; that is, if $\lambda$ and $\mu$ are partitions of $n$ and $\lambda$ is majorized by $\mu$, then $\mu^{*}$ is majorized by $\lambda^{*}$

Khoobchandra Agrawal

Numerade Educator

01:06

Problem 29

Prove that the number of partitions of the positive integer $n$ into parts each of which is at most 2 equals $\lfloor n / 2\rfloor+1$. (Remark: There is a formula, namely the nearest integer to $\frac{(n+3)^{2}}{12}$, for the number of partitions of $n$ into parts each of which is at most 3 but it is much more difficult to prove. There is also one for partitions with no part more than 4, but it is even more complicated and difficult to prove.)

Wendi Zhao

Numerade Educator

03:00

Problem 30

Prove that the partition function satisfies
$$p_{n}>p_{n-1} \quad(n \geq 2)$$

Mitchell Riley

Numerade Educator

01:55

Problem 31

Evaluate $h_{k-1}^{(k)}$, the number of regions into which $k$ -dimensional space is partitioned by $k-1$ hyperplanes in general position.

Audrey Fong

Numerade Educator

01:16

Problem 32

Use the recurrence relation (8.31) to compute the small Schröder numbers $s_{8}$ and $s_{9}$.

Prashansha Kaushik

Numerade Educator

01:16

Problem 33

Use the recurrence relation $(8.32)$ to compute the large Schr?der numbers $R_{7}$ and $R_{8}$. Verify that $R_{7}=2 s_{8}$ and $R_{8}=2 s_{9}$, as stated in Corollary 8.5.8.

Prashansha Kaushik

Numerade Educator

01:00

Problem 34

Use the generating function for the large Schröder numbers to compute the first few large Schröder numbers.

Harsh Gadhiya

Numerade Educator

02:40

Problem 35

Use the generating function for the small Schr?der numbers to compute the first few small Schröder numbers.

Bryan Lynn

Numerade Educator

01:26

Problem 36

. Prove that the Catalan number $C_{n}$ equals the number of lattice paths from $(0,0)$ to $(2 n, 0)$ using only upsteps $(1,1)$ and downsteps $(1,-1)$ that never go above the horizontal axis (so there are as many upsteps as there are downsteps). (These are sometimes called Dyck paths.)

Clarissa Noh

Numerade Educator

07:21

Problem 37

The large Schröder number $C_{n}$ counts the number of subdiagonal HVD-lattice paths from $(0,0)$ to $(n, n) .$ The small Schröder number counts the number of dissections of a convex polygonal region of $n+1 .$ Since $R_{n}=2 s_{n+1}$ for $n \geq 1$, there are as many subdiagonal HVD-lattice paths from $(0,0)$ to $(n, n)$ as there are dissections of a convex polygonal region of $n+1$ sides. Find a one-to-one correspondence between these lattice paths and these dissections.

Chris Trentman

Numerade Educator