species-0.1: Combinatorial species library

Math.Combinatorics.Species.Unlabelled

Description

An interpretation of species as ordinary generating functions, which count unlabelled structures.

Synopsis

# Documentation

Extract the coefficients of an ordinary generating function as a list of Integers. In particular, `unlabelled s !! n` is the number of unlabelled s-structures on an underlying set of size n. For example:

``` > take 10 \$ unlabelled octopi
[0,1,2,3,5,7,13,19,35,59]
```

gives the number of unlabelled octopi on 0, 1, 2, 3, ... 9 elements.

Actually, the above is something of a white lie, as you may have already realized by looking at the input type of `unlabelled`, which is `SpeciesAlg` rather than the expected `GF`. The reason is that although products and sums of unlabelled species correspond to products and sums of ordinary generating functions, composition and differentiation do not! In order to compute an ordinary generating function for a species defined in terms of composition and/or differentiation, we must compute the cycle index series for the species and then convert it to an ordinary generating function. So `unlabelled` actually works by first reifying the species to an AST and checking whether it uses composition or differentiation, and using operations on cycle index series if it does, and (much faster) operations directly on ordinary generating functions otherwise.