Coursera – Analytic Combinatorics: Part I + Part II (Princeton University)

This course teaches a calculus that enables precise quantitative predictions of large combinatorial structures. Part I covers generating functions and real asymptotics and then introduces the symbolic method in the context of applications in the analysis of algorithms and basic structures such as permutations, trees, strings, words, and mappings.

Analytic Combinatorics aims to enable precise quantitative predictions of the properties of large combinatorial structures. The theory has emerged over recent decades as essential both for the scientific analysis of algorithms in computer science and for the study of scientific models in many other disciplines, including probability theory, statistical physics, computational biology and information theory. Part I of this course covers recurrence relations, generating functions, asymptotics, and fundamental structures such as trees, permutations, strings, tries, words, and mappings, in the context of applications to the analysis of algorithms.

Part 1:

Lecture 1 Analysis of Algorithms
Lecture 2 Recurrences
Lecture 3 Solving recurrences with GFs
Lecture 4 Asymptotics
Lecture 5 The symbolic method
Lecture 6 Trees
Lecture 7 Permutations
Lecture 8 Strings and Tries
Lecture 9 Words and Mappings

Part2:

Lecture 1 Combinatorial Structures and OGFs
Lecture 2 Labelled Structures and EGFs
Lecture 3 Combinatorial Parameters and MGFs
Lecture 4 Complex Analysis, Rational and Meromorphic Asymptotics
Lecture 5 Applications of Rational and Meromorphic Asymptotics
Lecture 6 Singularity Analysis of Generating Functions
Lecture 7 Applications of Singularity Analysis
Lecture 8 Saddle-Point Asymptotics