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 Möbius function and Möbius algebra, Lagrange inversion formula, the exponential formula and tree enumeration.

The material has applications to active areas of research including polytopal theory, hyperplane arrangements, computational commutative algebra, representation theory and symmetric functions.

Prerequisite: A graduate course in linear algebra or permission of instructor.

COURSE OUTLINE

• Generating Functions.
• Stirling Numbers of the First and Second Kind.
• Permutations and Permutation Statistics.
• q-analogues.
• The Twelvefold Way.
• Principle of Inclusion-Exclusion.
• Partially Ordered Sets and Lattices.
• The Fundamental Theorem of Distributive Lattices.
• The Incidence Algebra.
• The Möbius Inversion Formula.
• The Möbius Function and Computational Techniques.
• The Möbius Algebra.
• Semi-modular Lattices and Hyperplane Arrangements.
• The Zeta Polynomial.
• Rank-selection.
• R-labelings.
• Eulerian Posets.
• Exponential Generating Functions.
• The Exponential Formula.
• Tree Enumeration.
• Lagrange Inversion Formula.

REFERENCES
Richard P. Stanley, Enumerative combinatorics. Vol. 2. Cambridge Studies in Advanced Mathematics, 62. Cambridge University Press, Cambridge, 1999.

Herb Wilf, Generatingfunctionology, Second edition, Academic Press, Inc., Boston, MA, 1994.