CombinatoricsTopics - Topics for Qualifying Exam in...

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I. Enumeration A. Selections with and without repetitions (combinations & permutations) B. Partitions 1. Stirling numbers of the first and second kind C. Principle of Inclusion-Exclusion 1. Surjections 2. Derangements 3. Euler ± function D. Generating functions (ordinary and exponential) 1. For combinations and permutations 2. Solutions of recurrence relations 3. Catalan numbers E. Pólya theory II. Combinatorial designs A. Design parameters B. Balanced incomplete block designs (BIBDs) C. Partially-balanced incomplete balanced designs (PBIBDs) D. Triple systems E. Resolvable designs F. Finite geometries 1. Affine geometries 2. Projective geometries 3. Partial geometries G. Latin squares 1. Orthogonality III. Matroids A. Independence, basis, spanning, and cycle systems; rank function and closure operator 1. The definition in terms of each of these 2. Proofs of equivalences of these definitions B. Duality C. Representability D. Greedy algorithm IV. Pigeonhole principle and Ramsey's theorem
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CombinatoricsTopics - Topics for Qualifying Exam in...

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