2. Teaching
3. MAT526

Welcome

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

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.