# Category Archives: combinatorics

## Efficiently listing orthobraces

In my last couple posts, we talked about a simple yet inefficient method for listing all orthobraces of a particular size. So how do we generate them efficiently? It turns out that it can be done: in 2011, Karim, Sawada, … Continue reading

Posted in combinatorics, computation | | 1 Comment

## Haskell code to naively list orthobraces

Let’s see some simple Haskell code to generate orthobraces, by generating all sequences and throwing away ones we’ve already generated. First, some library imports we’ll need. > import Data.List > import qualified Data.Set as S Here’s a function to generate … Continue reading

Posted in combinatorics, computation | | 1 Comment

## Listing orthobraces

I took a bit of a break to finish writing a paper for submission to the International Conference on Functional Programming—which I should write about here! All in good time. (I tend to accumulate things to write about faster than … Continue reading

Posted in combinatorics, computation | | 2 Comments

## Orthogons and orthobraces

One of these days soon I will get back to writing about primality tests, but for now I am having fun getting sidetracked on orthogons! In a previous post I gave rules for when two orthogons will be considered the … Continue reading

Posted in combinatorics, geometry | | 4 Comments

## Properties of orthogons II

In my previous post I proved three out of the four properties of orthogons I originally stated. Now let’s prove the final property: Every sequence of an even number of X’s and V’s, with exactly four more X’s than V’s, … Continue reading

Posted in combinatorics, geometry | | 3 Comments

## Properties of orthogons I

First things first: from now on, when talking about polygons with only right angles, instead of calling them “orthogonal polygons” I’m going to start calling them “orthogons”, which sounds cool, is much less clunky than “orthogonal polygons”, and doesn’t seem … Continue reading

Posted in combinatorics, geometry, proof | | 10 Comments

## Orthogonal polygons

It’s time to say more about PWW #21, in which I exhibited things like this: Quite a few commenters figured out what was going on, and mentioned several nice (equivalent) ways to think about it. Primarily, the idea is to … Continue reading

Posted in combinatorics, geometry | Tagged , , | 5 Comments

## Post without words #21

Posted in combinatorics, geometry, posts without words | Tagged , , | 17 Comments

## Fermat’s Little Theorem: proof by necklaces

It’s time for our second proof of Fermat’s Little Theorem, this time using a proof by necklaces. As you know, proof by necklaces is a very standard technique for… wait, what do you mean, you’ve never heard of proof by … Continue reading

Posted in combinatorics, number theory, primes, proof | | 4 Comments

## Möbius inversion

In my last post we saw that , that is, the Möbius function is the inverse of with respect to Dirichlet convolution. This directly leads to an interesting principle called Möbius inversion. Möbius inversion. Suppose is defined for as the … Continue reading

Posted in combinatorics, proof | Tagged , , , , | 3 Comments