Enumeration under group action: Unsolved graphical enumeration problems, IV

Abstract

This review article presents three methods for solving enumeration problems which can be construed as the determination of the number of orbits of an appropriate permutation group. Such a group must be constructed in accordance with the idiosyncrasies of the configurations to be counted and the equivalence relation on them. Thus three different binary operations on permutation groups, the sum, product, and power group, are defined and the structure of each is investigated. Applications of the corresponding theorems for enumeration under group action are provided. We conclude with a table of 27 current unsolved problems in graphical enumeration.