# Proofs that Really Count: The Art of Combinatorial Proof

Arthur T. Benjamin
Jennifer J. Quinn
Volume: 27
Edition: 1
Pages: 209
https://www.jstor.org/stable/10.4169/j.ctt6wpwjh

1. Front Matter
(pp. i-viii)
2. Foreword
(pp. ix-xii)

Every proof in this book is ultimately reduced to a counting problem| typically enumerated in two different ways. Counting leads to beautiful, often elementary, and very concrete proofs. While not necessarily the simplest approach, it offers another method to gain understanding of mathematical truths. To a combinatorialist, this kind of proof is the only right one. We offerProofs That Really Countas the counting equivalent of the visual approach taken by Roger Nelsen inProofs Without Words I & II[37, 38].

As human beings we learn to count from a very early age. A typical 2 year old...

(pp. xiii-xiv)
4. CHAPTER 1 Fibonacci Identities
(pp. 1-16)

Definition TheFibonacci numbersare defined by F0= 0, F1= 1, and for$n \ge 2$, Fn= Fn−1+ Fn−2.

The first few numbers in the sequence of Fibonacci numbers are 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, . . ...

How many sequences of 1s and 2s sum to n? Let’s call the answer to this counting question fn. For example, f4=5 since 4 can be created in the following 5 ways:

1 + 1 + 1 + 1, 1 + 1 + 2, 1 + 2 + 1, 2 + 1 + 1, 2 + 2....

5. CHAPTER 2 Gibonacci and Lucas Identities
(pp. 17-34)

Definition TheGibonacci numbers${G_n}$are defined by nonnegative integers${G_0},{G_1}$and for$n \ge 2,{G_n} - {G_{n - 1}} + {G_{n - 2}}$.

Definition TheLucas numbers${L_n}$are defined by${L_0} = 2,{L_1} = 1$and for$n \ge 2,{L_n} = {L_{n - 1}} + {L_{n - 2}}$.

The first few numbers in the sequence of Lucas numbers are 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199,....

In this chapter, we pursue identities involvingGibonacci numbers,which is shorthand for generalized Fibonacci numbers. There are many ways to generalize the Fibonacci numbers, and we shall pursue many of these generalizations in the next chapter, but for our purposes, we say a sequence of nonnegative integers${G_0},{G_1},{G_2},...$is a...

6. CHAPTER 3 Linear Recurrences
(pp. 35-48)

Definition Given integers${c_1},...,{c_k}$, akth order linear recurrenceis defined by${a_0},{a_1},...,{a_{k - 1}}$, and for$n > k,$,${a_n} = {c_1}{a_{n - 1}} + {c_2}{a_{n - 2}} + \cdot \cdot \cdot + {c_k}{a_{n - k}}$.

Definition Given integerssandt,theLucas sequence of the first kindis defined by${U_0} = 0,{U_1} = 1$and for$n > 2$,${U_n} = s{U_{n - 1}} + t{U_{n - 2}}$. For combinatorial convenience, we also define for$n \ge - 1$,${u_n} = {U_{n + 1}}$. Whens=t= 1, these are the Fibonacci numbers:${U_n} = {F_n}$, and${U_n} = {F_n}$.

Definition Given integerssandt,theLucas sequence of the second kindis defined by${V_0} = 2,{V_1} = s$and for${V_n} = s{V_{n - 1}} + t{V_{n - 1}} + t{V_{n - 2}}$. When$s = t = 1$, these are the

Lucas numbers:${V_n} = {V_n}$.

The recurrence for Fibonacci numbers can be extended in many different...

7. CHAPTER 4 Continued Fractions
(pp. 49-62)

Definition Given integers${a_0} \ge 0,{a_1} \ge 1,{a_2} \ge 1,...,{a_n} \ge 1$, define$[{a_0},{a_1},...{a_n}]$to be the fraction in lowest terms for

$a{}_0 - \frac{1}{{{a_1} + \frac{1}{{{a_2} + \frac{1}{{ \ddots - \frac{1}{{{a_n}}}}}}}}}$.

For example,$[2,3,4] = \frac{{30}}{{13}}$

You might be surprised to learn that the finite continued fraction

$3 + \frac{1}{{7 + \frac{1}{{15 + \frac{1}{{1 + \frac{1}{{292}}}}}}}}$and its reversal$292 + \frac{1}{{1 + \frac{1}{{15 + \frac{1}{{7 + \frac{1}{3}}}}}}}$

have the same numerator. These fractions simplify to$\frac{{103993}}{{33102}}$and$\frac{{103993}}{{355}}$respectively. In this chapter, we provide a combinatorial interpretation for the numerators and denominators of continued fractions which makes this reversal phenomenon easy to see. Our interpretation also allows us to visualize many important identities involving continued fractions.

First, we define some basic terminology. Given an infinite sequence of integers${a_0} \ge 0,{a_1} \ge 1,{a_2} \ge 1$, . . . let...

8. CHAPTER 5 Binomial Identities
(pp. 63-80)

Definition Thebinomial coefficient$\left( \begin{array}{l} n \\ k \\ \end{array} \right)$is the number ofk-element subsets of {1, . . . , n}.

Definition Themultichoose coefficient($\left( \begin{array}{l} n \\ k \\ \end{array} \right)$) is the number ofk-element subsets of {1, . . . , n}.

Examples of binomial coefficients are$\left( {\begin{array}{*{20}{c}} 4 \\ 0 \\ \end{array}} \right) = 1$,$\left( {\begin{array}{*{20}{c}} 4 \\ 1 \\ \end{array}} \right) = 4$,$\left( {\begin{array}{*{20}{c}} 4 \\ 2 \\ \end{array}} \right) = 6$,$\left( {\begin{array}{*{20}{c}} 4 \\ 3 \\ \end{array}} \right) = 4$, and$\left( {\begin{array}{*{20}{c}} 4 \\ 4 \\ \end{array}} \right) = 1$.

Examples of multichoose coefficients are ($\left( {\begin{array}{*{20}{c}} 4 \\ 0 \\ \end{array}} \right) = 1$,$\left( {\begin{array}{*{20}{c}} 4 \\ 1 \\ \end{array}} \right) = 4$,$\left( {\begin{array}{*{20}{c}} 4 \\ 2 \\ \end{array}} \right) = 6$,$\left( {\begin{array}{*{20}{c}} 4 \\ 3 \\ \end{array}} \right) = 4$, and$\left( {\begin{array}{*{20}{c}} 4 \\ 4 \\ \end{array}} \right) = 35$.

Binomial coefficients were born to count! Unlike most of the quantities we have discussed in this book, binomial coefficients are almost always defined as the answer to a counting problem. Specifically, we define$\left( \begin{array}{l} n \\ k \\ \end{array} \right)$to be the number of...

9. CHAPTER 6 Alternating Sign Binomial Identities
(pp. 81-90)

In the last chapter, we proved, for$n > 0$,$\sum\nolimits_{k > 0} {\left( {\begin{array}{*{20}{c}} n \\ {2k} \\ \end{array}} \right) = {2^{n - 1}}}$. Since$\sum\nolimits_{k \ge 0} {\left( {\begin{array}{*{20}{c}} n \\ {2k} \\ \end{array}} \right) = {2^n}}$, this implies that half of all subsets of$\left\{ {1, \ldots ,n} \right\}$are even. Consequently

$\sum\limits_{k \ge 0} {\left( {\begin{array}{*{20}{c}} n \\ {2k} \\ \end{array}} \right)} = \sum\limits_{k \ge 0} {\left( {\begin{array}{*{20}{c}} n \\ {2k + 1} \\ \end{array}} \right)}$.

This suggests that there should be a simple one-to-one correspondence between the even subsets of$\left\{ {1,2, \ldots ,n} \right\}$and the odd ones. We begin with a bijective proof of this fact.

Identity 167 For$n > 0$,

$\sum\limits_{k = 0}^n {\left( {\begin{array}{*{20}{c}} n \\ k \\ \end{array}} \right)} {( - 1)^k} = 0$.

Set 1: Let$\varepsilon$denote the set of even subsets$\left\{ {{a_1}, \ldots ,{a_k}} \right\}$of$\left\{ {1, \ldots ,n} \right\}$wherekis an even number. This set has size${\sum _k}$even$\left( \begin{array}{l} n \\ k \\ \end{array} \right)$.

Set 2: Let$o$denote the set of odd subsets$\left\{ {{a_1}, \ldots ,{a_k}} \right\}$of$\left\{ {1, \ldots ,n} \right\}$wherekis an odd number. This set has size${\sum _k}$even$\left( \begin{array}{l} n \\ k \\ \end{array} \right)$.

Correspondence: For...

10. CHAPTER 7 Harmonic and Stirling Number Identities
(pp. 91-108)

Definition Thenth harmonic numberis${H_n} = 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n}$. The first few harmonic numbers are${H_1} = 1$,${H_2} = \frac{3}{2}$,${H_3} = \frac{11}{6}$,${H_4} = \frac{25}{12}$.

Definition TheStirling number of the first kind$\left[ {\begin{array}{*{20}{c}} n \\ k \\ \end{array}} \right]$counts the number of permutations ofnelements with exactlykcycles. Some examples when$k = 2:\left[ {\begin{array}{*{20}{c}} 4 \\ 0 \\ \end{array}} \right] = 0$,$\left[ {\begin{array}{*{20}{c}} 2 \\ 2 \\ \end{array}} \right] = 1$,$\left[ {\begin{array}{*{20}{c}} 3 \\ 2 \\ \end{array}} \right] = 3$,$\left[ {\begin{array}{*{20}{c}} 4 \\ 2 \\ \end{array}} \right] = 11$,$\left[ {\begin{array}{*{20}{c}} 5 \\ 2 \\ \end{array}} \right] = 50$.

Definition TheStirling number of the second kind$\left[ {\begin{array}{*{20}{c}} n \\ k \\ \end{array}} \right]$counts the number of partitions ofnelements with exactlyksubsets. Some examples when$k = 2:\left[ {\begin{array}{*{20}{c}} 4 \\ 0 \\ \end{array}} \right] = 0$,$\left[ {\begin{array}{*{20}{c}} 2 \\ 2 \\ \end{array}} \right] = 1$,$\left[ {\begin{array}{*{20}{c}} 3 \\ 2 \\ \end{array}} \right] = 3$,$\left[ {\begin{array}{*{20}{c}} 4 \\ 2 \\ \end{array}} \right] = 7$,$\left[ {\begin{array}{*{20}{c}} 5 \\ 2 \\ \end{array}} \right] = 15$.

Harmonic numbers are defined to be partial sums of the harmonic series. That is, for...

11. CHAPTER 8 Number Theory
(pp. 109-124)

In this chapter, we have collected identities from arithmetic, algebra and number theory.

What could be simpler than the sum of the firstnnumbers? You probably already know this first identity. In fact, it’s a special case of Identity 135 in Chapter 5. You may recall that

$\sum\limits_{k = 1}^n k = \frac{{n(n + 1)}}{2}$.

Combinatorially, this can be rewritten and explained in two different ways. The subsequent identities and their counting proofs hinge on the interpretation of$\frac{{n(n + 1)}}{2}$as$\left( {\frac{{n + 1}}{2}} \right)$, a selection without repetition, or ($\left( {\frac{{n }}{2}} \right)$), a selection with repetition.

Identity 211 For$n \ge 0$,

$\sum\limits_{k = 1}^n k = \left( {\frac{{n + 1}}{2}} \right)$.

Question: How many ways can two different numbers be selected from the set {0, 1, . . . , n}?...

12. CHAPTER 9 Advanced Fibonacci & Lucas Identities
(pp. 125-146)

We end this book as we began it, by exploring more Fibonacci and Lucas identities (Lucas might say that we’ve come “full circle”!) We include some of the proofs that we found particularly challenging. As a climax, we add a dose of probability to obtain combinatorial proofs of theBinet formulas

${F_n} = \frac{1}{{\sqrt 5 }}\left[ {{{\left( {\frac{{1 + \sqrt 5 }}{2}} \right)}^n} - {{\left( {\frac{{1 - \sqrt 5 }}{2}} \right)}^n}} \right]$

and

${L_n} = {\left( {\frac{{1 + \sqrt 5 }}{2}} \right)^n} + {\left( {\frac{{1 - \sqrt 5 }}{2}} \right)^n}$

We conclude with some known identities that, as far as we know, have not yet succumbed to combinatorial interpretation.

Recall by Combinatorial Theorem 1 that fnis the number of square and domino tilings of a lengthnboard. The first identity is a warm-up to...

13. Some Hints and Solutions for Chapter Exercises
(pp. 147-170)
14. Appendix of Combinatorial Theorems
(pp. 171-172)
15. Appendix of Identities
(pp. 173-186)
16. Bibliography
(pp. 187-190)
17. Index
(pp. 191-193)