In
combinatorial
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many app ...
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the theory of combinatorial species is an abstract, systematic method for deriving the
generating functions
In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary series ...
of discrete structures, which allows one to not merely count these structures but give
bijective proofs involving them. Examples of combinatorial species are (
finite
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marked ...
)
graphs
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discre ...
,
permutation
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or proc ...
s,
trees
In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, including only woody plants with secondary growth, plants that are u ...
, 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, carefully elaborated and applied by the Canadian group of people around
André Joyal
André Joyal (; born 1943) is a professor of mathematics at the Université du Québec à Montréal who works on category theory. He was a member of the School of Mathematics at the Institute for Advanced Study in 2013, where he was invited to jo ...
.
The power of the theory comes from its level of abstraction. The "description format" of a structure (such as
adjacency list
In graph theory and computer science, an adjacency list is a collection of unordered lists used to represent a finite graph. Each unordered list within an adjacency list describes the set of neighbors of a particular vertex in the graph. This is ...
versus
adjacency matrix
In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph. The elements of the matrix indicate whether pairs of vertices are adjacent or not in the graph.
In the special case of a finite simp ...
for graphs) is irrelevant, because species are purely algebraic.
Category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
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.
The category of species is equivalent to the category of
symmetric sequence In algebraic topology, an \mathbb-object (also called a symmetric sequence) is a sequence \ of objects such that each X(n) comes with an actionAn action of a group ''G'' on an object ''X'' in a category ''C'' is a functor from ''G'' viewed as a cate ...
s in finite sets.
Definition of species
Any structure — an instance of a particular species — is associated with some
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
, and there are often many possible structures for the same set. For example, it is possible to construct several different graphs whose node labels are drawn from the same given set. At the same time, any set could be used to build the structures. The difference between one species and another is that they build a different set of structures out of the same base set.
This leads to the formal definition of a ''combinatorial species''. Let
be the
category
Category, plural categories, may refer to:
Philosophy and general uses
* Categorization, categories in cognitive science, information science and generally
*Category of being
* ''Categories'' (Aristotle)
*Category (Kant)
*Categories (Peirce)
* ...
of
finite set
In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example,
:\
is a finite set with five elements. Th ...
s, with the
morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
s of the category being the
bijection
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
s between these sets. A ''species'' is a
functor
In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
:
For each finite set ''A'' in
, the finite set ''F''
'A''ref group=note>Joyal prefers to write