Combinatorial Species - Definition of Species

Definition of Species

Any structure — an instance of a particular species — is associated with some set, 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 of finite sets and bijections (the collection of all finite sets, and invertible functions between them). A species is a functor

given a set A, it yields the set F of F-structures on A. A functor also operates on bijections. If φ is a bijection between sets A and B, then F is a bijection between the sets of F-structures F and F, called transport of F-structures along φ.

For example, the "species of permutations" maps each finite set A to the set of all permutations of A, and each bijection from A to another set B naturally induces a bijection from the set of all permutations of A to the set of all permutations of B. Similarly, the "species of partitions" can be defined by assigning to each finite set the set of all its partitions, and the "power set species" assigns to each finite set its power set.

There is a standard way of illustrating an instance of any structure, regardless of its nature. The diagram below shows a structure on a set of five elements: arcs connect the structure (red) to the elements (blue) from which it is built.

The choice of as the category on which species operate is important. Because a bijection can only exist between two sets when they have the same size, the number of elements in F depends only on the size of A. (This follows from the formal definition of a functor.) Restriction to finite sets means that |F| is always finite, so it is possible to do arithmetic with such quantities. In particular, the exponential generating series F(x) of a species F can be defined:

where is the size of F for any set A having n elements.

Some examples:

• The species of sets (traditionally called E, from the French "ensemble", meaning "set") is the functor which maps A to {A}. Then, so .
• The species S of permutations, described above, has . .
• The species T₂ of pairs (2-tuples) is the functor taking a set A to A2. Then and .