generatingfunctionology, analytic combinatorics and the kernel method
It is not common to put analytic and combinatorics together. However, generating functions are widely used in the study of enumerative combinatorics. The idea of analytic combinatorics is to solve the discrete problems through the analysis of the generating functions. Generating functions can be treated both as algebraic objects (formal series ring) and continuous functions. This will lead to many special and interesting techniques.
The aim of this course is to introduce basics ideas and some recent achievements of analytic combinatorics. So we will start from the introduction of generating functions and then pick some topics in analytic combinatorics. Many concepts, although not relevant to combinatorics in general, will appears in this course.
The aim of this course is to introduce basics ideas and some recent achievements of analytic combinatorics. So we will start from the introduction of generating functions and then pick some topics in analytic combinatorics. Many concepts, although not relevant to combinatorics in general, will appears in this course.
Lecturer
Date
15th October, 2025 ~ 5th January, 2026
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Monday,Wednesday | 09:50 - 11:25 | A3-2-303 | ZOOM 11 | 435 529 7909 | BIMSA |
Prerequisite
complex analysis, calculus, undergraduate algebra, some basic knowledge of algebraic curve
Syllabus
1 calculations of generating functions.
2 snake oil methods, Sieve methods.
3 Symbolic methods of unlabelled and labelled class.
4 the kernel method and basic analytic combinatorics of direct lattice path.
Then, depending on time, I will pick some of the following topics:
5 quarter plane lattice walk.
6 Riemann boundary value problem with Carleman shift.
7 Tutte invariants.
8 polynomial equation with one catalytic variable.
9 pattern avoiding permutations
10 hook-length formula
11 Potts model, chromatic numbers
2 snake oil methods, Sieve methods.
3 Symbolic methods of unlabelled and labelled class.
4 the kernel method and basic analytic combinatorics of direct lattice path.
Then, depending on time, I will pick some of the following topics:
5 quarter plane lattice walk.
6 Riemann boundary value problem with Carleman shift.
7 Tutte invariants.
8 polynomial equation with one catalytic variable.
9 pattern avoiding permutations
10 hook-length formula
11 Potts model, chromatic numbers
Reference
Wilf, Herbert S. generatingfunctionology. CRC press, 2005.
Flajolet, Philippe, and Robert Sedgewick. Analytic combinatorics. cambridge University press, 2009.
Bousquet-Mélou, Mireille, and Arnaud Jehanne. "Polynomial equations with one catalytic variable, algebraic series and map enumeration." Journal of Combinatorial Theory, Series B 96.5 (2006): 623-672.
Banderier, Cyril, and Philippe Flajolet. "Basic analytic combinatorics of directed lattice paths." Theoretical Computer Science 281.1-2 (2002): 37-80.
Fayolle, Guy, et al. Random walks in the quarter-plane. Vol. 40. New York: Springer-Verlag, 1999.
Bousquet-Mélou, Mireille, and Marni Mishna. "Walks with small steps in the quarter plane." Algorithmic probability and combinatorics 520 (2010): 1-39.
Bernardi, Olivier, and Mireille Bousquet-Mélou. "Counting colored planar maps: algebraicity results." Journal of Combinatorial Theory, Series B 101.5 (2011): 315-377.
Bernardi, Olivier, and Mireille Bousquet-Mélou. "Counting coloured planar maps: differential equations." Communications in Mathematical Physics 354.1 (2017): 31-84.
Melczer, Stephen. An Invitation to Analytic Combinatorics. Springer, 2021.
Flajolet, Philippe, and Robert Sedgewick. Analytic combinatorics. cambridge University press, 2009.
Bousquet-Mélou, Mireille, and Arnaud Jehanne. "Polynomial equations with one catalytic variable, algebraic series and map enumeration." Journal of Combinatorial Theory, Series B 96.5 (2006): 623-672.
Banderier, Cyril, and Philippe Flajolet. "Basic analytic combinatorics of directed lattice paths." Theoretical Computer Science 281.1-2 (2002): 37-80.
Fayolle, Guy, et al. Random walks in the quarter-plane. Vol. 40. New York: Springer-Verlag, 1999.
Bousquet-Mélou, Mireille, and Marni Mishna. "Walks with small steps in the quarter plane." Algorithmic probability and combinatorics 520 (2010): 1-39.
Bernardi, Olivier, and Mireille Bousquet-Mélou. "Counting colored planar maps: algebraicity results." Journal of Combinatorial Theory, Series B 101.5 (2011): 315-377.
Bernardi, Olivier, and Mireille Bousquet-Mélou. "Counting coloured planar maps: differential equations." Communications in Mathematical Physics 354.1 (2017): 31-84.
Melczer, Stephen. An Invitation to Analytic Combinatorics. Springer, 2021.
Audience
Advanced Undergraduate
, Graduate
, Postdoc
, Researcher
Video Public
Yes
Notes Public
Yes
Language
English