Basic Combinatorics, Spring 2015 Department of Mathematics, University of Notre Dame Instructor: David Galvin Abstract This document includes • homeworks, quizzes and exams, and • some supplementary notes from the Spring 2015 incarnation of Math 40210 | Basic Combinatorics, an undergraduate course offered by the Department of Mathematics at the University of Notre Dame. The notes have been written in a single pass, and as such may well contain typographical (and sometimes more substantial) errors. Comments and corrections will be happy received at [email protected]. Contents 1 General information about homework 3 2 Homework 1 (due January 23) 4 3 Homework 2 (due January 30) 12 4 Quiz 1 (February 4) 20 5 Quiz 1 Solutions 21 6 Homework 3 (due February 13) 22 7 Homework 3 Solutions 30 8 Homework 4 (due February 27) 34 9 Homework 4 Solutions 42 10 Quiz 2 (February 25) 46 11 Quiz 2 Solutions 47 12 Midsemester exam (March 4) 48 13 Homework 5 (due March 20) 54 14 Graph terminology 59 15 Homework 6 (due April 1) 61 1 16 Quiz 3 (March 30) 66 17 Homework 7 (due April 20) 67 18 Homework 8 (due April 29) 71 2 1 General information about homework In general homework will be assigned on a Friday, and will be due in class the following Friday. As they are assigned, homework problems will be added to this document, and will be posted on the website. At the same time I will make an email announcement. I will distribute a printout of the homework problems in class each week, with plenty of blank space. The solutions that you submit should be presented on this printout (using the flip side of a page if necessary). You may also print out the relevant pages from the website; if you do so, you must staple your pages together. I reserve the right to • not grade any homework that is not presented on the printout, and • deduct points for homework that is not stapled. The weekly homework is an important part of the course; it gives you a chance to think more deeply about the material, and to go from seeing (in lectures) to doing. It's also your opportunity to show me that you are engaging with the course topics. Homework is an essential part of your learning in this course, so please take it very seriously. It is extremely important that you keep up with the homework, as if you do not, you may quickly fall behind in class and find yourself at a disadvantage during exams. You should treat the homework as a learning opportunity, rather than something you need to get out of the way. Reread, revise, and polish your solutions until they are correct, concise, efficient, and elegant. This will really deepen your understanding of the material. I encourage you to talk with your colleagues about homework problems, but your final write-up must be your own work. Homework solutions should be complete (and in particular presented in complete sentences), with all significant steps justified. I reserves the right to • not grade any homework that is disorganized and incoherent. 3 2 Homework 1 (due January 23) Name: The purpose of this homework is to get used to using the pigeon-hole principle and the method of induction. Reading: Chapters 1 and 2 of the textbook. 1. 17 points are placed in an equilateral triangle of side length 1. Prove that two of the points are within distance 1=4 of each other. 4 2. (a) Select n + 1 distinct whole numbers between 1 and 2n. Prove that two of the numbers are coprime. (Numbers a and b are coprime if the greatest common divisor of a and b is 1.) (b) Is the conclusion still true if instead n numbers are selected? 5 3. For each n, exhibit a sequence of n2 real numbers that has neither a monotone increasing subsequence of length n + 1, nor a monotone decreasing subsequence of length n + 1 (this shows the the bound in the Erd}os-Szekeres theorem that we saw in class is tight). 6 4. Select 10 distinct two digit numbers. Prove that is is possible to select disjoint subsets A, B of the set of chosen numbers, so that the sum of the elements of A equals the sum of the elements of B. 7 5. Prove (by induction on n) that for all n ≥ 1, 1 1 1 p 1 + p + p + ::: + p ≤ 2 n: 2 3 n p (One can also prove by induction that the sum is at least 2( n + 1 − 1), and so get a very Pn −1=2 good estimate for i=1 i .) 8 6. The sum of the first n perfect kth powers turns out to be a polynomial in n of degree k + 1. For example, n(n + 1) 1 + 2 + 3 + ::: + n = 2 n(n + 1)(2n + 1) 12 + 22 + 32 + ::: + n2 = 6 n(n + 1)2 13 + 23 + 33 + ::: + n3 = : 2 Proof the second of these identities (12 + 22 + 32 + ::: + n2 = :::) by induction on n. 9 7. You are given a 64 by 64 chessboard, and 1365 L-shaped tiles (2 by 2 tiles with one square removed). One of the squares of the chessboard is painted purple. Is it possible to tile the chessboard using the given tiles, leaving only the purple square exposed? 10 8. In class we defined the number U(n) to be the number of different ways that people can be seated in a row of n adjacent chairs in an \unfriendly" way: no two adjacent chairs are occupied. We gave the following recursive specification of U(n): U(1) = 2, U(2) = 3 and n U(n) = Up(n−1)+U(n−2) for n ≥ 3, and we proved by induction that U(n) ≤ 1:25φ , where φ = (1 + 5)=2. Given an inductive proof of the following near-matching lower bound: there is a constant C > 0 (as part of your solution, you should figure out an explicit value for C) such that for all n ≥ 1, U(n) ≥ Cφn. 11 3 Homework 2 (due January 30) Name: The purpose of this homework is to become comfortable with the six basic counting problems introduced in lectures. Reading: Chapter 3 of the textbook. 1. (a) How many subsets does [n] have, that contain none of the elements 1; 2? (The notation \[n]" is shorthand for \f1; 2; 3; : : : ; ng".) (b) How many subsets does [n] have, that contain at least one of the elements 1; 2? (c) How many subsets does [n] have, that contain both of the elements 1; 2? 12 2. In how many ways can the elements of [n] be permuted (arranged in order), in such a way that 1 is listed later than both 2 and 3? 13 a1 a2 ak 3. Let n = p1 p2 : : : pk be the prime factorization of n (so each of the pi's are distinct prime numbers, p1 < p2 < : : : < pk, and each ai is a positive (non-zero) whole number). How many positive factors (including 1 and n) does n have? 14 4. Andy and Barbara play a game: they roll four dice. If at least one of the dice comes up six, Andy wins the game, and otherwise Barbara does. (a) In how many different ways can the game proceed (assuming the dice are distinguishable from each other)? (b) In how many of these ways does Andy win? (c) In how many of these ways does Barbara win? 15 5. In how many different ways can n rooks be placed on an n-by-n chessboard, with no two rooks attacking each other? (Two rooks attack each other if they are either in the same row as each other, or the same column). 16 6. My class has n sophomores, n juniors and n seniors. In how many ways can the class break up into n triples, with each triple including one sophomore, one junior and one senior? (The order of people within each group doesn't matter, nor does the order in which the groups are created.) 17 7. I invite n couples to a party. I want to ask some subset (perhaps empty) of the 2n attendees to give a speech, but I don't want to ask both members of any couple. In how many ways can I do this? [Your answer might be quite simple, or it might be a slightly less simple sum of binomial coefficients.] 18 8. For all n ≥ 1, the number of even-sized subsets of f1; : : : ; ng is equal to the number of odd- sized subsets of f1; : : : ; ng. Proof this bijectively. Meaning: let En be the set of even-sized subsets of f1; : : : ; ng, and let On be the set of odd-sized subsets of f1; : : : ; ng. Give a bijection f : En !On. Don't forget to prove that the function you give is both injective and surjective. 19 4 Quiz 1 (February 4) Name: 1. How many different anagrams does the word MISSISSIPPI have? (The four S's are indistinguishable from one another, as are the four I's and two P's.) 2. Give a combinatorial proof of the binomial coefficient identity n n − 1 k = n ; k k − 1 by describing a counting problem whose answer is both the left- and right-hand side of the above equality, depending on how you count. Clearly describe both ways of counting. 20 5 Quiz 1 Solutions 1. How many different anagrams does the word MISSISSIPPI have? (The four S's are indistinguishable from one another, as are the four I's and two P's.) 11! Solution: 4!4!2! .