Math 566: Combinatorial Theory

Winter 2017

Course meets: Tuesday and Thursday 1:10-2:30 in 4096 East Hall.

Instructor: Sergey Fomin, 4868 East Hall, 764-6297, fomin@umich.edu

Office hours: Tuesday and Thursday 2:40-4:00 in 4868 East Hall.

Grader: Qingzhong Liang, qzliang@umich.edu.

Course homepage: http://www.math.lsa.umich.edu/~fomin/566w17.html

Level: introductory graduate/advanced undergraduate.

Prerequisites: Familiarity with formal proofs, and with basic notions of combinatorics. Linear algebra will be used throughout.

Student work expected: several problem sets.

Synopsis: This course is an introduction to algebraic and enumerative combinatorics at the beginning graduate level. Topics include: fundamentals of algebraic graph theory; applications of linear algebra to enumeration of matchings, tilings, and spanning trees; combinatorics of electric networks; partially ordered sets; integer partitions and Young tableaux.

Optional text: R. P. Stanley, Algebraic Combinatorics: Walks, Trees, Tableaux, and More, Springer, 2013. The text of this book (without exercises) is available at the link above.

Topics covered (tentative list, subject to change):

ALGEBRAIC GRAPH THEORY Graph eigenvalues Enumeration of walks Domino tilings Random spanning trees Resistor networks Matrix-tree theorem Cayley's Theorem Eulerian tours De Bruijn Sequences PARTITIONS AND TABLEAUX Partitions and Young diagrams Pentagonal Number Theorem q-binomial coefficients Sperner's theorem Unimodality of Gaussian coefficients Hooklength formula Schensted correspondence Tableaux and involutions Knuth equivalence and Greene's theorem