Analytic Combinatorics

Lecturer: 　CHEUNG, Yun Kuen (Marco)

Origin: During Feb-May 2026, I had the pleasure of teaching the Math Special Topics Course "Combinatorics, Probability and Algorithms" at ANU. While the course touched on many facets of the three areas in its title, we did not have sufficient time to do justice to Analytic Combinatorics — a subject that reveals the elegant connection between the asymptotic behaviors of combinatorial structures and the complex analysis of generating functions. I have compiled this lecture series to provide a comprehensive introduction, and I hope this can help you appreciate the beauty of the subject as much as I do.

Prerequisite: 　Mathematical knowledge in the following subjects is deemed necessary:

• Enumerative Combinatorics: permutations, combinations, inclusion-exclusion principle, Möbius inversion
• Analysis: limit of sequence \(\lim_{n\rightarrow\infty} a_n\), radius of convergence of \(f(z) = \sum_{n=0}^\infty a_n z^n\), asymptotics such as \(\Theta(n^{n+1/2}e^{-n})\) and \(n!\sim \sqrt{2\pi n} (\frac ne)^n\)
• Probability: expected values of discrete and continuous probability distribution
• Linear Algebra: arithmetics of vectors and matrices like computing \(\vec{y}^\textsf{T} \mathbf{A} \vec{x} \), solving linear systems \(\mathbf{A} \vec{x} = \vec{b}\), Cramer's rule
• Calculus: partial derivatives \(\frac{\partial f}{\partial x_i}\), integration \(\int_a^b f(x)\,\mathsf{d}x\)

You will benefit more from this course if you have already done complex analysis (analytic function, Cauchy integral).

Lecture	Latest Update	Note
Lecture 1: Symbolic Method for Unlabelled Structures and Ordinary Generating Functions	2026-07-10
Lecture 2: Advanced Unlabelled Constructions — Multisets, Power Sets and Cycles	2026-07-10
Lecture 3: Symbolic Method for Labelled Structures and Exponential Generating Functions	2026-07-13
Lecture 4: More Applications of Labelled Constructions and Boxed Product	2026-07-16
Lecture 5: Functional Equations and Lagrange Inversion	2026-07-23