User:Richard Pinch/sandbox-17

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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 $$ R(x) = \sum_{n=0}^\infty |R[n]|\, \frac{x^n}{n!} $$

Some special examples. $$ \mathbf{0} : V \mapsto \emptyset $$ $$ \mathbf{1} : V \mapsto \left\lbrace{ \begin{array}{cl} \{\emptyset\} & \text{if } V = \emptyset \\ \emptyset & \text{otherwise} \end{array} }\right. $$ $$ \mathbf{X} : V \mapsto \left\lbrace{ \begin{array}{cl} \{v\} & \text{if } V = \{v\} \\ \emptyset & \text{otherwise} \end{array} }\right. $$

The generating functions of these species are $0$, $1$ and $x$ respectively. A species is connected if $|R[1]| = 1$.

A natural transformation between species $H \rightarrow R$ is a family of maps $H[V] \rightarrow R[V]$ compatible with relabelling.

Further examples. Lists and permutations both have generating function $\frac{1}{1-x}$. They are not isomorphic.

Operations. The sum of two species $A+B$ is the disjoint union $(A+B)[V] = A[V] \sqcup B[V]$. The product of two species is given by a sum over partitions $$ (A.B)[V] = \sum_{V = V_1 \sqcup V_2} A[V_1] \times B[V_2] $$

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