Math 128

Current Problems in Combinatorics

Last updated June 17, 2015 12:56:59 EDT

## Tentative syllabus

This is a tentative list of topics that I plan to cover, with references to the main sources that I will use.

- Pattern-avoiding permutations.

[Vince Vatter's notes given in class]
- Motivation. Sorting using stacks and queues.
- Closed classes of permutations. Separable permutations.
- Asymptotics and growth rates.
- The Stanley-Wilf Conjecture and the Marcus-Tardos proof.

[Bona, *Combinatorics of permutations*, Chapman Hall-CRC, 2004]
- Exact enumeration of patterns of length 4.
- Packing densities.

[Elizalde, Deutsch, A simple and unusual bijection for Dyck paths and its consequences, *Ann. Comb.* 7 (2003), 281–297]

[Elizalde, Pak, Bijections for refined restricted permutations, *J. Combin. Theory Ser. A* 105 (2004), 207-219]
- Statistics on pattern-avoiding permutations.

- Generating trees, functional equations, and the kernel method.

[West, Generating trees and the Catalan and Schröder numbers, *Discrete Math.* 146 (1995), 247–262]

[Bousquet-Melou, Four Classes of Pattern-Avoiding Permutations Under One Roof: Generating Trees with Two Labels, *Electron. J. Combin.* 9 (2002-3), #19]

[Flajolet, Sedgewick, *Analytic Combinatorics*, Section VII.8.1]

[Bousquet-Melou, Mishna, Walks with small steps in the quarter plane, *Contemp. Math.* 520 (2010), 1-40]
- Constructing generating trees with one label.
- From generating trees to functional equations.
- The kernel method for generating trees with one label, and for one-dimensional walks.
- Generating trees with two labels.
- Examples of the kernel method for generating trees with two labels: Baxter permutations, 1234-avoiding permutations.
- Walks with small steps in the quarter plane.
- Generalized paterns.

[Babson, Steingrimsson, Generalized permutation patterns and a classification of the Mahonian statistics, *Sem. Loth. Comb.* 44 (2000), B44b]

[Steingrimsson, Generalized permutation patterns - a short survey, in *Permutation Patterns*, LMS Lecture Notes Cambridge Univ. Press (2010), 137–152]

[Claesson, Generalized pattern avoidance, *Europ. J. Combinatorics* 22 (2001), 961-971]

- Student presentations.

Last updated June 17, 2015 12:56:59 EDT