Category Archives: species

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

Posted on February 13, 2018 by Brent

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 | Tagged , , , , , , | 6 Comments

Signed sets and ballots and naturality

Posted on February 27, 2017 by Brent

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 | Tagged , , , , , , , | 2 Comments

Signed sets and ballots, part 2

Posted on February 24, 2017 by Brent

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 | Tagged , , , , , , , | 1 Comment

Signed sets and ballots, part 1

Posted on February 20, 2017 by Brent

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 | Tagged , , , , , , | 2 Comments

Boltzmann sampling for generic Arbitrary instances

Posted on March 23, 2016 by Brent

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 →

Posted in combinatorics, haskell, math, species | Tagged , , , , , | 14 Comments

The Species of Bracelets

Posted on July 10, 2015 by Brent

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 →

Posted in combinatorics, haskell, math, species | Tagged , , | 5 Comments

Crystal Ball Connection Patterns via Species and Generating Functions

Posted on June 2, 2015 by Brent

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 | Tagged , , , , , | 1 Comment

Pan-Galactic Division in Haskell

Posted on May 12, 2015 by Brent

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 →

Posted in haskell, math, species | Tagged , , , , , , , , , | 6 Comments

Readers wanted!

Posted on August 10, 2014 by Brent

tl;dr: Read a draft of my thesis and send me your feedback by September 9! Over the past year I’ve had several people say things along the lines of, “let me know if you want me to read through your … Continue reading →

Posted in category theory, combinatorics, diagrams, grad school, math, species, writing | Tagged , , , , | 15 Comments

Anafunctors

Posted on June 16, 2014 by Brent

This is part four in a series of posts on avoiding the axiom of choice (part one, part two, part three). In my previous post, we considered the “Axiom of Protoequivalence”—that is, the statement that every fully faithful, essentially surjective … Continue reading →

Posted in category theory, math, species | Tagged , , , , , , , , , , | 5 Comments