Algebraic Combinatorics
Via Finite Group Actions

A. Betten, H. Fripertinger, A. Kerber

August 28, 2001

  • 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
  • Constructions
  • Orbit evaluation
  • Transversals of symmetry classes
  • Orbits of centralizers
  • Recursion and orderly generation
  • Generating orbit representatives
  • Symmetry adapted bases
  • Index
    • harald.fripertinger@kfunigraz.ac.at,
    last changed: August 28, 2001