Welcome to the course web page for the Fall 2016 manifestation of MAT 526: Topics in Combinatorics at Northern Arizona University. We will be using the recently published textbook Eulerian Numbers by T. Petersen (DePaul University).

AMB 176

10:15-11:15 MWF and 9-10 TTh (or by appointment)

dana.ernst@nau.edu

928.523.6852

dcernst.github.io/teaching/mat526f16

The tentative plan is to cover Chapters 1-6 and 11 of Eulerian Numbers, but we may cover more or less depending on time and interests. Here are the proposed topics:

- Eulerian numbers
- Binomial coefficients
- Generating functions
- Classical Eulerian numbers
- Eulerian polynomials
- Two important identities
- Exponential generating function
- Narayana numbers
- Catalan numbers
- Pattern-avoiding permutations
- Narayana numbers
- Dyck paths
- Planar binary trees
- Noncrossing partitions
- Partially ordered sets
- Basic definitions and terminology
- Labeled posets and P-partitions
- The shard intersection order
- The lattice of noncrossing partitions
- Absolute order and Noncrossing partitions
- Gamma-nonnegativity
- The idea of gamma-nonnegativity
- Gamma-nonnegativity for Eulerian numbers
- Gamma-nonnegativity for Narayana numbers
- Palindromicity, unimodality, and the gamma basis
- Computing the gamma vector
- Real roots and log-concavity
- Symmetric boolean decomposition
- Weak order, hyperplane arrangements, and the Tamari lattice
- Inversions
- The weak order
- The braid arrangement
- Euclidean hyperplane arrangements
- Products of faces and the weak order on chambers
- Set compositions
- The Tamari lattice
- Rooted planar trees and faces of the associahedron
- Refined enumeration
- The idea of a $q$-analogue
- Lattice paths by area
- Lattice paths by major index
- Euler-Mahonian distributions
- Descents and major index
- $q$-Catalan numbers
- $q$-Narayana numbers
- Dyck paths by area
- Coxeter groups
- The symmetric group
- Finite Coxeter groups: generators and relations
- $W$-Mahonian distribution
- $W$-Eulerian numbers
- Finite reflection groups and root systems
- The Coxeter arrangement and the Coxeter complex
- Action of $W$ and cosets of parabolic subgroups
- Counting faces in the Coxeter complex
- The $W$-Euler-Mahonian distribution
- The weak order
- The shard intersection order

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MAT 441: Topology

MAT 526: Combinatorics

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