Topics on Combinatorics, Algebra and Geometry
TU Graz, Winter Term 2023-2024

Lecturer: Cesar Ceballos

Times and location

Lecture: Friday 8:00-9:30 Seminarraum 2 (Geometrie) (NT04064)
Exercises: Friday 9:45-10:30 Seminarraum 2 (Geometrie) (NT04064)

Course description

The contents of this class is the interesting interplay betwen algebra, geometry and combinatorics which occurs in subjects such as polytope theory, Hopf algebras, diagonal harmonics, and others.

The course is offered to master and PhD students as part of the module ''Grundthemen Diskrete Mathematik''.


This will be regularly updated to include references to the topics of each lecture.

[BB1] Anders Bjorner and Francesco Brenti, Combinatorics of Coxeter Groups, Graduate Texts in Mathematics, 231. Springer-Verlag Berlin Heidelberg, 2005.
[Hag1] James Haglund, The q,t-Catalan Numbers and the Space of Diagonal Harmonics, University Lecture Series, 41. American Mathematical Society, Providence, RI, 2008.
[Hum1] James E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge studies in advanced mathematics, 29. Cambridge University Press, 1990.
[Sta1] Richard Stanley, Enumerative Combinatorics, Volume 1. Second edition. Cambridge Stud. Adv. Math., 49 Cambridge University Press, 2012.
[Zie1] Günter Ziegler, Lectures on Polytopes, Graduate Texts in Mathematics, 152. Springer-Verlag, New York, 1995.

Articles: (more or less in the order they are used in the lectures)
[CG1] Cesar Ceballos, Rafael S. González D'León, Signature Catalan combinatorics, J. Comb.10 (2019), no.4, 725-773.
[Rot1] Gian-Carlo Rota, On the foundations of combinatorial theory I. Theory of Möbius Functions, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2 (1964), 340-368.
[H1] M. D. Haiman, Conjectures on the quotient ring by diagonal invariants, J. Algebraic Combin. 3:1 (1994), 17-76.
[CGH1] Cesar Ceballos, Tom Denton, and Christopher R. H. Hanusa, Combinatorics of the zeta map on rational Dyck paths, J. Combin. Theory Ser. A 141 (2016), 33-77.
[A1] Jaclyn Anderson, Partitions which are simultaneously t1- and t2-core, Discrete Math. 248 (2002) 237-243.
[AHJ1] Drew Armstrong, Christopher R.H. Hanusa, Brant C. Jones, Results and conjectures on simultaneous core partitions, European J. Combin. 41 (2014) 205-220.
[ALW1] D. Armstrong, N. A. Loehr, and G. S. Warrington, Sweep maps: A continuous family of sorting algorithms, Adv. Math. 284 (2015), 159-185.
[ALW1] H. Thomas and N. Williams, Sweeping up zeta, Selecta Math. 24 (2018), no. 3, 2003-2034.
[Ed1] Paul H. Edelman, Chain enumeration and noncrossing partitions, Discrete Math. 31 (1980), no. 2, 171-180.
[Ed2] Paul H. Edelman, Multichains, noncrossing partitions and trees, Discrete Math. 40 (1982), no. 2-3, 171-179.
[CSZ1] Cesar Ceballos, Francisco Santos and Günter M. Ziegler, Many non-equivalent realizations of the associahedron, Combinatorica 35 (2015), no. 5, 513-551.

Tentative Schedule

(this is not final, it will be updated weekly)

Date Lecture notes Exercises Contents
Oct. 6 Lecture 0 Vorbeschprechung
Oct. 13 Lecture 1 Exercise Sheet 1 Three combinatorial sequences:
Factorial numbers, Catalan numbers, and parking functions.
Enumeration, combinatorial models and bijections.
Oct. 20 Lecture 2 Exercise Sheet 2 q-analogs
The space of harmonics
Oct. 27 Lecture cancelled (Fenstertag).
Nov. 3 Lecture 3 Exercise Sheet 3 The space of diagonal harmonics
Nov. 10 Lecture cancelled.
Nov. 17 Lecture 4 Exercise Sheet 4 q,t-Catalan combinatorics
area, dinv, bounce statistics
zeta map
Nov. 24 Lecture 5 Exercise Sheet 5 Rational q,t-Catalan combinatorics
simultaneous (a,b)-core partitions
rational (a,b)-Dyck paths
skew length statistic
Dec. 1 Lecture 6 Exercise Sheet 6 The rational zeta map
Dec. 15 Lecture 7 Exercise Sheet 7 Partially ordered sets, lattices
Incident Algebra, Möbius inversion
Jan. 12 Lecture 8 Exercise Sheet 8 The weak order on permutations
The Tamari lattice
The lattice of partitions
The lattice of noncrossing partitions
Jan. 19 Lecture 9 No exercises The lattice of noncrossing partitions:
multichains and Möbius function
The permutahedron
Jan. 26 Lecture 10 No exercises The associahedron

Holidays and University Closings: Dec. 8 (Maria Empfängnis), Dec. 21-Jan. 6 (Weihnachtsferien).


There are two options for the final grade:
- a final oral exam about the contents of the course, or
- a written final project on a selected topic related to the course (individually or in small groups 2-3 people), together with an oral presentation about it. In this case, the student should have shown active participation in the lecture and dominance of the topics of the course through an active presentation of exercises.

Potential final projects

Project A. The lattice of non-crossing partitions, maximal chains and Möbius function
Project B. The volume of permutahedra and parking functions
Project C. The n! and the (n+1)^{n-1} conjectures/theorems.
Project D. The rational zeta map and area preserving involutions.
Project E. Rational combinatorics and rational zeta map for Coxeter groups
Project F. The lattice of noncrossing partitions for Coxeter Coxeter groups
Project G. Cambrian lattices
Project H. Subword complexes, Brick polytopes and brick polyhedra
Project I. Flow polytopes, associahedra and permutahedra
Project J. The Pitman-Stanley polytope and parking functions
Project K. The diameter of the associahedron