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CO330ExtraProbs

# CO330ExtraProbs - C&O 330 EXTRA PROBLEMS COMBINATORIAL...

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Unformatted text preview: C&O 330 EXTRA PROBLEMS COMBINATORIAL ENUMERATION INSTRUCTOR: D.M.JACKSON I have collected together some questions for review purposes. Many you will have seen before, but you should try to do them without looking at any other material - at least in the first place. A few of them will be new to you. Here are some tips for reviewing the material of the course. In the lec- tures I have covered partitions of integers and sets, the Pattern Algebra, the Maximal Decomposition Theorem, P´olya’s Theorem, the theory of ex- ponential generating series and simple applications of P´olya’s Theorem, and a substantial number of examples of the use of this material. These were the main topics. (1) Proofs (a) You should understand the proofs of Lagrange’s Implicit Func- tion Theorem and P´olya’s, but I shall not ask you to prove them. (b) You should be able to prove results such as the Permutation Lemma and the Maximal Decomposition Theorem. (2) Using the theory (a) You should be able to use the theory of partitions, the theory of exponential generating series, the Pattern Algebra, the Max- imal Decomposition Theorem and Lagrange’s Implicit Function Theorem and P´olya’s Theorem to solve enumerative questions of the sort seen in the course. (b) I shall not be asking you questions that involve the sophisticated double or reverse uses of Lagrange’s Theorem that we saw in a few identities. (c) You should be able to provide a good level of explanation in your solutions. Such explanations are a vital part of your solutions, and you will be given credit for them. (d) You should understand the main combinatorial decompositions for permutations, partitions of sets, trees, functions, matrices, partitions of integers, Ferrers diagrams and so on. 1 2 INSTRUCTOR: D.M.JACKSON (3) You should reread the solutions for the assignments that I have placed on the website. You should be able to understand all of the details that are given there. (4) You should reread the relevant sections of the Course Notes that contain the material discussed in class. You will already be aware of the fact that these Notes contain more than I have covered in class. The additional material is there for people who wish to see some more complex applications of the material. Problems - NOT to be handed in (1) Find the generating series for the number of paths on the integer sublattice of the real plane that start at (0 , 0) and end at ( n,n ) , with steps that are unit line segments in the positive Ox or Oy direction, and that do not meet the line x = y at points with odd coordinates. (2) The partitions of 5 are 5 , 41 , 32 , 311 , 22 1 , 21 11 , 1 1 11 1 . The to- tal number of ”1”s in this set is 12. Now look at each partition one by one and count the number of distinct symbols in each par- tition. For the seven partitions of 5 listed above these numbers are 1 , 2 , 2 , 2 , 2 , 2 , 1 . The sum of these is again 12. Prove that this is true in general....
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CO330ExtraProbs - C&O 330 EXTRA PROBLEMS COMBINATORIAL...

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