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Difference between revisions of "User:Richard Pinch/sandbox-17"

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(Start article: Combinatorial species)
 
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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.   
 
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 $V$.   
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*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 $R[\{1,\ldots,n\}]$.
We write $R[n]$ for $R[\{1,\ldots,n\}]$.
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*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 $R[f\circ g] = R[f] \circ R[g]$.
*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]]
 
The ''(exponential) generating function'' of $R$ is the [[formal power series]]

Revision as of 15:30, 22 July 2021

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

How to Cite This Entry:
Richard Pinch/sandbox-17. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Richard_Pinch/sandbox-17&oldid=51761