A combinatorial species is a functor $F: \mathbf{B} \to \mathbf{B}$, where $\mathbf{B}$ is the category of finite sets and bijections. For each finite set $U$, $F[U]$ is the set of $F$-structures on $U$, and for each bijection $\sigma: U \to V$, $F[\sigma]: F[U] \to F[V]$ is a transport of structures that preserves composition and identity.

A species $F$ has three associated generating series:

• Type generating series: $\widetilde{F}(x) = \sum_{n \geq 0} f_n x^n$, where $f_n = |F[n]/S_n|$ (number of isomorphism types)
• Exponential generating series: $F(x) = \sum_{n \geq 0} |F[\{1,\ldots,n\}]| \frac{x^n}{n!}$
• Cycle index series: $Z_F = \sum_{n \geq 0} \frac{1}{n!} \sum_{\sigma \in S_n} |\operatorname{Fix}_{F[n]}(\sigma)| \cdot p_1^{c_1(\sigma)} \cdots p_n^{c_n(\sigma)}$

For species $F$ and $G$ (with $G[\emptyset] = \emptyset$), the substitution $F \circ G$ is defined by $(F \circ G)[U] = \bigsqcup_{\pi \in \operatorname{Par}(U)} F[\pi] \times \prod_{B \in \pi} G[B],$ where $\operatorname{Par}(U)$ is the set of partitions of $U$. This corresponds to plethystic substitution of cycle index series.

Singularity analysis provides asymptotic formulas for the coefficients of generating functions. If $f(z) \sim C(1 - z/\rho)^{-\alpha}$ as $z \to \rho$ (the dominant singularity), then $[z^n]f(z) \sim \frac{C}{\Gamma(\alpha)} n^{\alpha - 1} \rho^{-n}$. This transfer theorem, combined with species operations, enables automatic asymptotic enumeration.