The goal of this paper is to develop extensions of Polya enumeration methods which count orbits of functions. De Bruijn, Harary, and Palmer all worked on this problem and created generalizations which involve permuting the codomain and domain of functions simultaneously. We cover their results and specifically extend them to the case where the group of permutations need not be a direct product of groups. In this situation, we develop a way of breaking the orbits into subclasses based on a characteristic of the functions involved. Additionally, we develop a formula for the number of orbits made up of bijective functions. As a final extension, we also expand the set we are acting on to be the set of all relations between finite sets. Then we show how to count the orbits of relations.

The classification of finite semigroups is difficult even for small orders because of their large number. Most finite semigroups are nilpotent of nilpotency rank 3. Formulae for their number up to isomorphism, and up to isomorphism and anti-isomorphism of any order are the main results in the theoretical part of this thesis. Further studies concern the classification of nilpotent semigroups by rank, leading to a full classification for large ranks. In the computational part, a method to find and enumerate multiplication tables of semigroups and subclasses is presented. The approach combines the advantages of computer algebra and constraint satisfaction, to allow for an efficient and fast search. The problem of avoiding isomorphic and anti-isomorphic semigroups is dealt with by supporting standard methods from constraint satisfaction with structural knowledge about the semigroups under consideration. The approach is adapted to various problems, and realised using the computer algebra system GAP and the constraint solver Minion. New results include the numbers of semigroups of order 9, and of monoids and bands of order 10. Up to isomorphism and anti-isomorphism there are 52,989,400,714,478 semigroups with 9 elements, 52,991,253,973,742 monoids with 10 elements, and 7,033,090 bands with 10 elements. That constraint satisfaction can also be utilised for the analysis of algebraic objects is demonstrated by determining the automorphism groups of all semigroups with 9 elements. A classification of the semigroups of orders 1 to 8 is made available as a data library in form of the GAP package Smallsemi. Beyond the semigroups themselves a large amount of precomputed properties is contained in the library. The package as well as the code used to obtain the enumeration results are available on the attached DVD.