18.217
Combinatorial Theory: Enumerative Combinatorics
Fall 2021, MIT
Instructor:
Alex Postnikov
Grader:
Pakawut
(Pro) Jiradilok
Class meetings:
MWF 23 pm
Room 2190
Office hours: email for appointments
Webpage:
math.mit.edu/18.217/
Course description:
The content of 18.217 varies from year to year.
This year we plan to concentrate on enumerative methods in combinatorics.
We will (more or less) cover the material of the first volume of
Richard Stanley's
Enumerative Combinatorics and some parts of the
second volume.
We will also discuss applications of enumerative combinatorics to other
areas of mathematics,
such as representation theory, convex geometry, and algebraic geometry.
We will talk about generating functions, posets, partitions, permutations,
graphs, polytopes, hyperplane arrangements, matroids, Young diagrams,
symmetric functions, Schubert
polynomials, and other topics (as time allows).
Course Level:
The course should be accessible to first year graduate students.
Grading: Based on several problem sets.
Problem Sets:
Problem set 1
due Friday, October 15, 2021
Problem Set 2
due Friday, December 3, 2021
Lectures:

W 9/8/2021: Introduction: What is Enumerative Combinatorics? [EC1, Section 1.1]

F 9/10/2021: The Twelvefold Way.
Set partitions. The Bell numbers and the Striling numbers of 2nd
kind. Rook placements. [EC1, Section 1.4]

M 9/13/2021: The Twelvefold Way (cont'd): Compositions and integer partitions.
Formal power series. The exponential and the composition formulas.
[EC2, Section 5.1]

W 9/15/2021: The Lagrange inversion.
Applications: the Catalan numbers and Cayley's formula for labelled
trees.

F 9/17/2021: A combinatorial proof of the Lagrange inversion formula:
plane trees, compositions, Dyck paths, and depthfirst search.
[EC2, Sections 5.3, 5.4]

M 9/20/2021: Depthfirst search. Two bijecions between plane trees and Dyck
paths. The ballot numbers and the Narayana numbers.
Statistics on Dyck paths and plane trees: peaks, double descents, first
runs, ground bumps, leaves, degree of the root.

W 9/22/2021:
Catalan statistics and permutation statistics.
The Narayana numbers vs the Eulerian numbers.
Proof of the formula for the Narayana numbers.
[EC1, Section 1.2]

F 9/24/2021:
The ballot numbers and the reflection principle.
Labelled binary trees and plane trees: increasing binary trees,
left increasing binary trees trees, increasing plane trees,
childrenincreasing plane trees.

M 9/27/2021: Parking functions and labelled Dyck paths.

W 9/29/2021: Generalized parking functions. The braid arrangement
and the Shi arrangement. Introduction to convex polytopes.

F 10/01/2021: Polytopes (cont'd).
Simple polytopes. The fvector and the hvector.
The nonnegativity and symmetry of the hvector.
Example: The permutohedron. A relationship between the Stirling
numbers of the 2nd kind and the Eulerian numbers.
o
Recommended books:
This webpage will be updated periodically. All information related to the
course, including problem sets, will be posted here.
last updated: October 1, 2021