[...]... the fascinating area of Pattern Avoidance, discussed in Chapter 14 I am deeply appreciative for manuscript reading by my colleagues Andrew Vince, Neil White, and Aleksandr Vayner xi xii A Walk Through Combinatorics A significant part of the first edition was written during the summer of 2001, when I enjoyed the hospitality of my parents, Miklos and Katalin Bona, at the Lake Balaton in Hungary My gratitude... means that the remaining teams still played at least 18 games on average against other remaining teams Now iterate this procedure- look for a team from the remaining group that has only played nine games and omit it As the number of teams is finite, this elimination procedure has to come to an end The only way that can happen is that there will be a group of which we cannot 14 A Walk Through Combinatorics. .. decrease the average number of opponents Indeed, as we are only interested in the number of opponents played (and not games), we can assume that any two teams played each other at most once The 18-game-average means that all the m Division One teams together played 9m games as a game involves two teams Omitting T, we are left with m — 1 teams, who played a grand total of at least 9m — 9 games This means... so we have found 100 non-negative sums Now assume that a 50 + agg < 0 Then necessarily ai + a 2 H h a4 g + a 5 i -I h a g8 + aioo > 0 (1.1) In this case we claim that all sums ai + aioo are non-negative To see this, it suffices to show that the smallest of them, ai + aioo is non-negative And that is true as 0 > a 50 + a9 g > 049 + ags > 048 + a9 7geq• •• > a 2 + a 5 i, and therefore the left-hand side...x A Walk Through Combinatorics the end of the book, should be acquainted with the extremely intriguing questions that abound in these two areas We wrote this book as we believe that combinatorics, researching it, teaching it, learning it, is always fun We hope that at the end of the walk, readers will agree **** Exercises that are thought to be significantly harder than average are marked by... them had to be lo as he or she did not shake anyone's hand, and the other one had to be Yi as he or she had only one handshake, and that was with Ys- As spouses do not shake hands, this implies that the spouse of I7 is either lo or Yi However, lo is married to Ys, so Yi must be married to Y7 By a similar argument that the reader should be able to complete, YQ A Walk Through Fig 1.6 Combinatorics Y& and... grateful to my students who never stopped asking questions and showed which part of the material needed further explanation Some of the presented material was part of my own research, sometimes in collaboration I would like to say thanks to my co-authors, Andrew MacLennan, Bruce Sagan, Rodica Simion, Daniel Spielman, Geza Toth, and Dennis White I am also indebted to my former advisor, Richard Stanley,... 4 Now take such a round robin tournament, and replace the teams with the numbers ai, 02, • • • , aioo- So the fifty games of each round are replaced by fifty pairs of type ai + aj As each team plays in each round, the sum of the 100 numbers, or 50 pairs, in any given round is zero Therefore, at least one pair must have a non-negative sum in any given row, otherwise that row would have a negative sum... Sciacca for the cover page If you do not know why a book entitled "A Walk Through Combinatorics" has such a cover page, you may figure it out when reading Chapter 10 After the publication of the first edition in 2002, several mathematicians contributed lists of typographical errors to be corrected Particularly extensive lists were provided by Margaret Bayer, Richard Ehrenborg, John Hall, Hyeongkwan... boxes that together contain exactly one hundred balls (5) + Last year, the Division One basketball teams played against an average of eighteen different opponents Is it possible to find a group of teams so that each of them played against at least ten other teams of the group? (6) (a) The set M consists of nine positive integers, none of which has a prime divisor larger than six Prove that M has two . that when I was a student beginning to learn combinatorics there was a textbook available as attractive as Bona& apos;s. Students today are fortunate to be able to sample the treasures available. elemen- tary. An undergraduate student eager to do some original research has a good chance of making a worthwhile contribution in the area of pattern avoidance. Vll viii A Walk Through Combinatorics. with a Foreword by Richard Stanley A Walk Through Combinatorics An Introduction to Enumeration and Graph Theory