Lecture: | Friday 8:00-9:30 | Seminarraum 2 (Geometrie) (NT04064) |
Exercises: | Friday 9:45-10:30 | Seminarraum 2 (Geometrie) (NT04064) |
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.
Books:[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. |
[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. |
(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 q,t-analogs |
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 Polytopes: The permutahedron |
Jan. 26 | Lecture 10 | No exercises | The associahedron |
Holidays and University Closings: Dec. 8 (Maria Empfängnis), Dec. 21-Jan. 6 (Weihnachtsferien).
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 |