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Combinatorial species

A class of finite labelled stuctures closed under relabelling. A contravariant functor from the category $\mathcal B$ of finite sets and bijections to the category $\mathcal F$ of finite sets and functions. A species $R$ thus determines the following data.

  • For a finite set $V$, a finite set $R[V]$, thought of as the $R$-structures with labels in $V$.

We write $R[n]$ for $R[\{1,\ldots,n\}]$.

  • For a bijection $f: V \rightarrow W$, a map $R[f] : R[V] \rightarrow R[W]$, with the properties

that $R[\mathrm{id}_V] = \mathrm{id}_{R[V]}$ and $R[f\circ g] = R[f] \circ R[g]$.

The (exponential) generating function of $R$ is the formal power series $$ R(x) = \sum_{n=0}^\infty \frac{|R[n]|}{n!} x^n $$

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