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

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.

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

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

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MAT 431: Intro to Analysis

MAT 526: Combinatorics

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