A method for counting certain types of combinatorial structures under symmetry by using permutation groups and generating functions.
Combinatorial structures: Different types of combinatorial structures (such as graphs, trees, permutations, set partitions, etc.) that can be used in conjunction with Polya enumeration theorem.
Symmetry groups: Basic concepts related to group theory, including groups, group actions, conjugacy classes, and stabilizers.
Burnside's lemma: A simple enumeration technique that provides a way to count the number of orbits of a group action on a set.
Cycle index polynomial: A polynomial expression that describes the action of a finite group on a set of indeterminates or variables. It can be used in conjunction with Polya enumeration theorem for the enumeration of combinatorial structures.
Exponential generating functions: A generating function that encodes the coefficients of a sequence of exponential type. It can be used to derive the cycle index polynomial and for the application of Polya enumeration theorem.
Polya enumeration theorem: A powerful technique for counting the number of orbits of a group action on a set of combinatorial structures. The theorem is based on the computation of the cycle index polynomial of the group.
Applications of Polya enumeration: Use cases and problem-solving techniques that leverage the power of Polya enumeration theorem in various fields such as chemistry, physics, and computer science.
Examples and exercises: A series of problems that help illustrate the concepts covered in the topics mentioned above. These include problems on counting the number of graphs up to symmetry, the number of necklaces that can be formed using different types of beads, and many more.
Polya enumeration theorem for cyclic groups: This theorem deals with the fundamental principles of counting the number of objects distributed in a cyclic manner under the action of finite cyclic groups.
Polya enumeration theorem for dihedral groups: This theorem deals with the enumeration of objects distributed in a dihedral group, which is a group of symmetries of a regular polygon.
Polya enumeration theorem for abelian groups: This theorem is used to count objects that are subjected to the action of a finite abelian group. Here, the objects can be of different types, and apply the group action in different ways.
Polya enumeration theorem for permutation groups: This theorem is used to count the number of fixed point-free permutations in a permutation group.
Polya enumeration theorem applied to graphs: This theorem is used to find the number of non-isomorphic graphs under the action of a given group.
Polya enumeration theorem applied to polyominoes: This theorem is used to find the number of different polyominoes (shapes created by unit squares) under the action of a given group.
Polya enumeration theorem applied to chemical molecules: This theorem is used to count the number of isomers (different arrangements of atoms) of a chemical molecule under the action of a given group.