Algebraic Combinatorics
Lecture Schedule: 4:00pm to 5:30pm Mon
4:00pm  5:30pm Friday
Venue: HSB 266
Prerequisites: A good understanding of Linear algebra and basic group theory.
Course Info:
Objectives:
The course aims to reveal the fascinating interplay between Algebra, Combinatorics, and Graphs. It not only builds the fundamentals for students who plan to do PhD in Algebra/ Combinatorics / Graph theory and related topics, but also useful to students and researchers in other areas of science and engineering to which the methods of Algebra, Combinatorics, Graph theory may be applied.
Course Contents:
Eigenvalues and Walks on graphs. Radon transform and hypercubes. Sperner property, lattices and boolean algebra. Enumeration under group actions. Ferrer's diagram, Young tableaux and Matrix Tree theorem. Applications to Electrical networks, planar graphs. Introduction to combinatorial commutative algebra.
Text Books:
Richard P Stanley, Algebraic Combinatorics : Walks  Trees  Tableaux and More, Springer, 2013.
A preliminary copy of the book is available from Stanley's webpage.
Reference Books:
 Richard P Stanley, Enumerative Combinatorics  Volume 1, Springer. 2001

Richard P Stanley, Enumerative Combinatorics  Volume 2, Springer, 2001.

Combinatorial Commutative Algebra. Erza Miller and Bernd Sturmfels. Springer, 2005.

Rafael H Villarreal, Monomial Algebras, CRC Press, 2015.

R B Bapat, Graphs and Matrices, Springer, 2014.
Introduction to the course.
Graphs, Simple graphs, walks. Adjacency matrix. Walks and eigenvalues.
Examples and Exercises.
Open walks in $K_n$. A quick visit to ordinary and exponential generation functions. A powerful identity regarding sums of complex powers. Eigenvalues from walks.
Assignment: Here is the first assignment with a minor correction.
Walking on the Hypercube: The Finite Radon Transform.
More on the Finite Radon Transform. The eigenvectors of an arbitrary graph and its meaning. Examples using SAGE.
Species of combinatorial structures.
Cannonical sum representation of a species. Products of species and examples. Composition of Species. Examples.
Derivative of a species. Pointing operation.
Assignment: Here is the second assignment.
Vertebrates and Cayley's formula. More on generating functions.
Markov Chains and Random Walks. Perron's theorem for positive matrices.
Frobenius extension of Perron's theorem for irreducible nonnegative matrices.
Hitting time using Linear Algebra.
Partial Orders and graded Posets. Sperner property of set systems. Some results on ordermatchings.
The boolean algebra $B_n$ has sperner property. Another proof (combinatorial) given by Lubell.
Group action. The quotient poset $B_n/G$ is ranksymmetric and rank unimodal.
The quotient poset $B_n/G$ has Sperner property. Reading asmt: Applications to edge reconstruction conjecture.
Integer partitions, Young's lattice and Young Diagrams. Examples. Properties of poset $L(m,n)$.
The $q$analogue of binomial coefficient. Poset $L(m,n)$ is rankunimodal and Sperner.
LGV Lemma and applications.
Lecturer: Suresh Dara.
Applications of sperner property of $L(m,n)$.
TBA
Thanks must go to my friend Amri from whose page I stole the template.