18.217 Combinatorial Theory: Enumerative Combinatorics

Fall 2021, MIT

Instructor: Alex Postnikov
Grader: Pakawut (Pro) Jiradilok    
Class meetings: MWF 2-3 pm     Room 2-190
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:

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

  2. 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]

  3. 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]

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

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

  6. M 9/20/2021: Depth-first 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.

  7. 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]

  8. 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, children-increasing plane trees.

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

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

  11. F 10/01/2021: Polytopes (cont'd). Simple polytopes. The f-vector and the h-vector. The nonnegativity and symmetry of the h-vector. Example: The permutohedron. A relationship between the Stirling numbers of the 2nd kind and the Eulerian numbers.
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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