Welcome

Welcome to the course web page for the Spring 2021 manifestation of MAT 526: Topics in Combinatorics at Northern Arizona University.

Course Info

Title: MAT 526: Topics in Combinatorics
Semester: Spring 2021
Credits: 3
Section: 1
Time: 9:10-10:00AM MWF
Location: AMB 207

Instructor Info

  Dana C. Ernst, PhD
  AMB 176
  MWF 1:30-2:30PM, T 1:00-2:00PM via Zoom
  dana.ernst@nau.edu
  928.523.6852
  danaernst.com


The mathematician does not study pure mathematics because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.

What is This Course All About?

This course focuses on enumerative combinatorics with an introduction to generating function techniques. The tentative plan is to cover Chapters 1-6 and 11 of Eulerian Numbers by T. Kyle Petersen (DePaul University), 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.)

Don’t fear failure. Not failure, but low aim, is the crime. In great attempts it is glorious even to fail.



Dana C. Ernst

Mathematics & Teaching

  Northern Arizona University
  Flagstaff, AZ
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Current Courses

  MAT 431: Intro to Analysis
  MAT 526: Combinatorics

About This Site

  This website was created using GitHub Pages and Jekyll together with Twitter Bootstrap.

  Unless stated otherwise, content on this site is licensed under a Creative Commons Attribution-Share Alike 4.0 International License.

  The views expressed on this site are my own and are not necessarily shared by my employer Northern Arizona University.

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Land Acknowledgement

  Flagstaff and NAU sit at the base of the San Francisco Peaks, on homelands sacred to Native Americans throughout the region. The Peaks, which includes Humphreys Peak (12,633 feet), the highest point in Arizona, have religious significance to several Native American tribes. In particular, the Peaks form the Diné (Navajo) sacred mountain of the west, called Dook'o'oosłííd, which means "the summit that never melts". The Hopi name for the Peaks is Nuva'tukya'ovi, which translates to "place-of-snow-on-the-very-top". The land in the the area surrounding Flagstaff is the ancestral homeland of the Hopi, Ndee/Nnēē (Western Apache), Yavapai, A:shiwi (Zuni Pueblo), and Diné (Navajo). We honor their past, present, and future generations, who have lived here for millennia and will forever call this place home.