## 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
• 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

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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