Combinatorics Through Guided Discovery, 2004a

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ix will find a real sense of satisfaction when you see what you can figure out with concentrated thought. Besides, you can get hints! Some of the questions will appear to be trick questions, especially when you get the answer. They are not intended as trick questions at all. Instead they are designed so that they don’t tell you the answer in advance. For example the answer to a question that begins “How many...” might be “none.” Or there might be just one example (or even no examples) for a problem that asks you to find all examples of something. So when you read a question, unless it directly tells you what the answer is and asks you to show it is true, don’t expect the wording of the problem to suggest the answer. The book isn’t designed this way to be cruel. Rather, there is evidence that the more open-ended a question is, the more deeply you learn from working on it. If you do go on to do mathematics later in life, the problems that come to you from the real world or from exploring a mathematical topic are going to be open-ended problems because nobody will have done them before. Thus working on open-ended problems now should help to prepare you to do mathematics later on. You should try to write up answers to all the problems that you work on. If you claim something is true, you should explain why it is true; that is you should prove it. In some cases an idea is introduced before you have the tools to prove it, or the proof of something will add nothing to your understanding. In such problems there is a remark telling you not to bother with a proof. When you write up a problem, remember that the instructor has to be able to “get” your ideas and understand exactly what you are saying. Your instructor is going to choose some of your solutions to read carefully and give you detailed feedback on. When you get this feedback, you should think it over carefully and then write the solution again! You may be asked not to have someone else read your solutions to some of these problems until your instructor has. This is so that the instructor can offer help which is aimed at your needs. On other problems it is a good idea to seek feedback from other students. One of the best ways of learning to write clearly is to have someone point out to you where it is hard to figure out what you mean. The crucial thing is to make it clear to your reader that you really want to know where you may have left something out, made an unclear statement, or failed to support a statement with a proof. It is often very helpful to choose people who have not yet become an expert with the problems, as long as they realize it will help you most for them to tell you about places in your solutions they do not understand, even if they think it is their problem and not yours! As you work on a problem, think about why you are doing what you are doing. Is it helping you? If your current approach doesn’t feel right, try to see why. Is this a problem you can decompose into simpler problems? Can you see a way to make up a simple example, even a silly one, of what the problem is asking you to do? If a problem is asking you to do something for every value of an integer n, then what happens with simple values of n like 0, 1, and 2? Don’t worry about making mistakes; it is often finding mistakes that leads mathematicians to their best insights. Above all, don’t worry if you can’t do a problem. Some problems are given as soon as there is one technique you’ve learned that might help do that problem. Later on there may be other techniques that you can bring back to that problem to try again. The notes have been designed this way on purpose. If
x you happen to get a hard problem with the bare minimum of tools, you will have accomplished much. As you go along, you will see your ideas appearing again later in other problems. On the other hand, if you don’t get the problem the first time through, it will be nagging at you as you work on other things, and when you see the idea for an old problem in new work, you will know you are learning. There are quite a few concepts that are developed in this book. Since most of the intellectual content is in the problems, it is natural that definitions of concepts will often be within problems. When you come across an unfamiliar term in a problem, it is likely it was defined earlier. Look it up in the index, and with luck (hopefully no luck will really be needed!) you will be able to find the definition. Above all, this book is dedicated to the principle that doing mathematics is fun. As long as you know that some of the problems are going to require more than one attempt before you hit on the main idea, you can relax and enjoy your successes, knowing that as you work more and more problems and share more and more ideas, problems that seemed intractable at first become a source of satisfaction later on. The development of this book is supported by the National Science Foundation. An essential part of this support is an advisory board of faculty members from a wide variety of institutions who have made valuable contributions. They are Karen Collins, Wesleyan University, Marc Lipman, Indiana University/Purdue University, Fort Wayne, Elizabeth MacMahon, Lafayette College, Fred McMorris, Illinois Institute of Technology, Mark Miller, Marietta College, Rosa Orellana, Dartmouth College, Vic Reiner, University of Minnesota, and Lou Shapiro, Howard University. The overall design and most of the problems in the appendix on exponential generating functions are due to Professors Reiner and Shapiro. Any errors or confusing writing in that appendix are due to me! I believe the board has managed both to make the book more accessible and more interesting.
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194 E. GNU Free Documentation Licen
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200 Index equivalent, 112 cycle ind
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202 Index pair structure, 159 pair,
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