# Category Archives: species

## A (work in progress) translation of Joyal’s original paper on species

tl;dr: I’m working on an English translation, with additional commentary, of Joyal’s 1981 paper introducing the concept of combinatorial species. Collaboration and feedback welcome! Back when I was writing my PhD thesis on combinatorial species, I was aware that André … Continue reading

Posted in combinatorics, projects, species, writing | | 6 Comments

## Signed sets and ballots and naturality

This is blog post #3 in a series on signed sets and ballots (the previous posts are here and here). Naturality, isomorphism, and equipotence When are two species isomorphic? Since species are, by definition, functors , the obvious answer is … Continue reading

Posted in combinatorics, math, species | | 2 Comments

## Signed sets and ballots, part 2

Recall, from my previous post, that our goal is to find a combinatorial proof showing the correspondence between signed sets and signed ballots, where a signed set is just a set of elements, considered positive or negative according to the … Continue reading

Posted in combinatorics, math, species | | 1 Comment

## Signed sets and ballots, part 1

The other day, Anders Claesson wrote a very nice blog post explaining a more combinatorial way to understand multiplicative inverses of virtual species (as opposed to the rather algebraic way I explained it in my previous post). In the middle … Continue reading

Posted in combinatorics, math, species | | 2 Comments

## Boltzmann sampling for generic Arbitrary instances

Update, 7/17/2017: this now exists; see https://byorgey.wordpress.com/2016/09/20/the-generic-random-library-part-1-simple-generic-arbitrary-instances/ . tl;dr: I know how to generate random instances of data types in a generic way, and even have some old code that already does all the hard work, but won’t have time … Continue reading

## The Species of Bracelets

Summary: The species package now has support for bracelets, i.e. equivalence classes of lists up to rotation and reversal. I show some examples of their use and then explain the (very interesting!) mathematics behind their implementation. I recently released a … Continue reading

## Crystal Ball Connection Patterns via Species and Generating Functions

A couple weeks ago, Denise posted Puzzle: Crystal Ball Connection Patterns on her blog, Let’s Play Math. I had fun playing with it and thought I would demonstrate how to apply some high-powered combinatorial techniques to it (probably not what … Continue reading

Posted in combinatorics, haskell, math, species | | 1 Comment

Summary: given an injective function , it is possible to constructively “divide by ” to obtain an injection , as shown recently by Peter Doyle and Cecil Qiu and expounded by Richard Schwartz. Their algorithm is nontrivial to come up … Continue reading