In combinatorial mathematics, the theory of combinatorial species is an abstract, systematic method for analysing discrete structures in terms of generating functions. Examples of discrete structures are (finite) graphs, permutations, trees, and so on; each of these has an associated generating function which counts how many structures there are of a certain size. One goal of species theory is to be able to analyse complicated structures by describing them in terms of transformations and combinations of simpler structures. These operations correspond to equivalent manipulations of generating functions, so producing such functions for complicated structures is much easier than with other methods. The theory was introduced by André Joyal.
The power of the theory comes from its level of abstraction. The "description format" of a structure (such as adjacency list versus adjacency matrix for graphs) is irrelevant, because species are purely algebraic. Category theory provides a useful language for the concepts that arise here, but it is not necessary to understand categories before being able to work with species.
Read more about Combinatorial Species: Definition of Species, Calculus of Species, Types and Unlabelled Structures, Class of All Species, Generalizations, Software
Famous quotes containing the word species:
“As kings are begotten and born like other men, it is to be presumed that they are of the human species; and perhaps, had they the same education, they might prove like other men. But, flattered from their cradles, their hearts are corrupted, and their heads are turned, so that they seem to be a species by themselves.... Flattery cannot be too strong for them; drunk with it from their infancy, like old drinkers, they require dreams.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)