Algebraic Combinatorics
Via Finite Group Actions

A. Betten, H. Fripertinger, A. Kerber

January 19, 2005

• Actions
• Actions of groups
• Orbits
• Stabilizers
• Fixed points
• Examples
• Cosets
• The Cauchy-Frobenius Lemma
• The permutation character
• The Cauchy-Frobenius Lemma 2
• Similar Actions
• Exercises
• Bilateral classes, symmetry classes of mappings
• Double Cosets
• Action on k-subsets
• Sylows Theorem
• Products of Actions
• Paradigmatic Examples
• Paradigmatic Examples 2
• Exercises
• Finite symmetric groups
• Cycle decomposition
• Conjugation
• Conjugacy Classes
• Generators of the symmetric group
• Subgroups of Cyclic Groups
• Colourings of the n Gon
• The Sign
• Splitting Orbits
• Rothe diagram and inversions
• The Lehmer code and reduced decompositions
• Sn as a Coxeter Group
• The Exchange Lemma
• Exercises
• Complete monomial groups
• Centralizers of elements in finite symmetric groups
• Conjugacy classes in complete monomial groups
• Examples
• Execises
• Enumeration of symmetry classes
• Graphs
• The cycle type of the induced action on 2-subsets
• Some congruences
• Exercises
• The involution principle
• Selfcomplementary graphs
• Involutions
• The Involution Principle
• The Principle of Inclusion and Exclusion
• The Garsia-Milne bijection
• Exercises
• Special symmetry classes
• Injective symmetry classes
• Surjective symmetry classes
• Various combinatorial numbers
• Exercises
• Weights
• Enumeration by weight
• Exercises
• Cycle indicator polynomials
• Exercises
• Sums of cycle indicators, recursive methods
• A generalization
• The Decomposition Theorem
• Species
• Marks
• Counting by stabilizer class
• Constructions
• Orbit evaluation
• Transversals of symmetry classes
• Orbits of centralizers
• Recursion and orderly generation
• Generating orbit representatives
• Symmetry adapted bases
• Index