So what is a species?

Formally, a species is a functor from the category $N$ of finite sets to itself, where the morphisms in $N$ are bijections of sets.

Less formally, a species $S$ is an assignment, for each $n\ge 0$, of the set $[n]=\{1,2,\ldots,n\}$ to a finite set $S([n])$, along with a rule for “relabeling.” The relabeling rule then determines a unique finite set $S(A)$ for any given finite set $A$.

For instance, we can define the “species of cyclic permutations” $C$ by $$C([n])=\{\text{cyclic permutations of }1,2,\ldots,n\}.$$ Then for any $A=\{a_1,\ldots,a_n\}$ and any bijection $\phi:[n]\to A$, we have $$C(A)=\{\text{cyclic permutations of }a_1,a_2,\ldots,a_n\},$$ and there is a natural bijection $C([n])\to C(A)$ given by re-labeling the permutations according to $\phi$.

Species serve as an excellent organizational tool for manipulating generating functions. The exponential generating function of a species $S$ is the function $$\widetilde{S}(x)=\sum_{n=0}^\infty \frac{|S([n])|}{n!}x^n.$$ Recall that we can add, multiply, and sometimes compose exponential generating functions, and it is useful to define corresponding operations on species:

Addition of species: Given two species $S,T$, we define the species $S+T$ by $$(S+T)([n])=S([n])\sqcup T([n]),$$ with the relabeling rule naturally inherited from $S$ and $T$. It is easy to see that this corresponds to addition of exponential generating functions: $\widetilde{S+T}=\widetilde{S}+\widetilde{T}$.

Example: Let $E$ be the species of even permutations and $O$ the species of odd permutations. Let $P$ be the species of all permutations. Then $E+O=P$.

Multiplication of species: Given species $S$ and $T$, we define the species $S\cdot T$ as follows:
$(S\cdot T)([n])$ is the set of tuples $(A_1,A_2,\alpha,\beta)$ where $A_1$ and $A_2$ form a partition of $[n]$, and $\alpha\in S(A_1)$ and $\beta\in S(A_2)$. We also have $\widetilde{S\cdot T}=\widetilde{S}\cdot \widetilde{T}$.

Example: Every permutation can be decomposed into a product of even cycles and odd cycles, so if $A$
is the species of permutations having only even cycles and $B$ is the species of permutations having only odd cycles, then $A\cdot B=P$.

Composition of species: Given species $S$ and $T$ with $T(\emptyset)=\emptyset$, we define
$(S\odot T)([n])$ to be the set of tuples $(A_1,\ldots,A_k,\alpha_1,\ldots \alpha_k,\beta)$ where $A_1,\ldots,A_k$ form a partition of $[n]$, each $\alpha_i\in T(A_i)$, and $\beta\in S(\{A_1,\ldots, A_k\})$. This corresponds to composition of exponential generating functions: $\widetilde{S\odot T}=\widetilde{S}\circ \widetilde{T}$, which is only well-defined when $\widetilde{T}(0)=0$.

Example: Permutations can be uniquely written as a union of disjoint cycles. Each of the cycles has a $C$-structure and the cycles themselves have the trivial structure, so if $E$ is the “trivial species” sending each $[n]$ to $\{1\}$, then $E\odot C=P$.