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Enumerative Combinatorics: Course Outline (MAT-648)

Enumerative combinatorics. ... More generally, given an infinite collection of finite sets Si indexed by the natural numbers, enumerative combinatorics seeks to describe a counting function which counts the number of objects in Sn for each n.

Course Outline

Enumerative Combinatorics: An introduction to the basic notions and techniques in enumerative combinatorics. Topics include generating functions, principle of inclusion and exclusion, bijections, recurrence relations, partially ordered sets, the Mobius function and Mobius algebra, Lagrange inversion formula, the exponential formula and tree enumeration.  Stirling numbers of the first and second kind . Permutations and permutation statistics. q-analogues. The twelve-fold way . Principle of inclusion-exclusion. Partially ordered sets and lattices.The fundamental theorem of distributive lattices . The incidence algebra. The Mobius inversion formula. The M¨obius function and computational techniques. The Mobius algebra.Semi-modular lattices and hyperplane arrangements. The zeta polynomial. Rank-selection. R-labeling. Eulerian posets.. Exponential generating functions. The exponential formula. Tree enumeration Lagrange inversion formula.

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