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).

Course Info

Title: MAT 526: Topics in Combinatorics
Semester: Spring 2016
Credits: 3
Section: 1
Time: MWF at 11:30AM-12:20PM
Location: AMB 207

Instructor Info

  Dana C. Ernst, PhD
  AMB 176
  10:15-11:15 MWF and 9-10 TTh (or by appointment)

Proposed Topics

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
The Tamari lattice given by the weak order on $S_4(231)$. (Figure 5.12 in Eulerian Numbers by T. Petersen.)

Dana C. Ernst

Mathematics & Teaching

  Northern Arizona University
  Flagstaff, AZ
  Google Scholar
  Impact Story

Current Courses

  MAT 220: Math Reasoning
  MAT 411: Abstract Algebra

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