Math 511: Spring, 2023
Syllabus

We are relying mainly on the bookCombinatorics: The Rota Way by Catherine Yan, Gian-Carlo Rota, and Joseph P. S. Kung; with some additions in the lectures and corrections in the errata.

Here is a list of topics (the syllabus, in the dictionary sense). Dates are approximate.

1. (1/18-20) Counting the four types of set function S → X. Falling factorials (x)_n. Bases for polynomials and basis conversion between monomials and falling factorials. Partition lattice Π_n.
   Goal: Introduce essence of much combinatorics: counting techniques and formulas.
2. (1/23-27) Chapter 1, §§1.1-2. Posets. Partition lattice. Lattices: approaches as poset or algebra.
   Omit the topology. Treat the infinite cases lightly or skip them. Our class is mainly concerned with finite combinatorics.
3. (1/27-2/3) §1.3. Lattices. Boolean algebras.
4. (1/30-2/6) §1.4. Partitions.
   Omit everything after Lemma 1.4.2.
5. (2/1-3) §1.3. Distributive lattices; Birkhoff's theorem.
6. (2/8) §1.5. Relations., mainly their abstract properties. Matrix rank, with correct definition of the determinant. Closure operators: standard and poset.
   You can omit the relative sum.
7. (2/10-22) §§2.1-2. Matchings.
8. (2/24) §2.3. Matrices.
9. (2/26-3/1) §2.4. Submodular functions; independent matchings.
10. (3/6-10) §2.6. Birkhoff polytope of doubly stochastic matrices.
11. (3/13) §2.7. Gale-Ryser Theorem and proof of necessity.
12. (3/15-3/29) §3.1. Möbius function of a poset.
13. (3/31-4/12) §3.2. Poset theorems.
14. (4/12-4/14) §3.3. Sperner theory. (Omit Littlewood–Offord at the end of the section.)
15. (4/17-4/26) §3.5. Modular and geometric lattices. Plus: Chromatic polynomial of a graph. Regions of a hyperplane arrangement.
16. (4/28-5/1) §4.1. Generating functions.
17. (5/2-3) §4.2-3. Polynomial sequences of binomial type (lightly); umbral calculus (slightly).