Category Archives: combinatorics

A combinatorial proof: the story so far

In my last post I reintroduced this seemingly odd phenomenon: Start with consecutive integers and raise them all to the th power. Then repeatedly take pairwise differences (i.e. subtract the first from the second, and the second from the third, … Continue reading

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A combinatorial proof: reboot!

More than seven years ago I wrote about a curious phenomenon, which I found out about from Patrick Vennebush: if you start with a sequence of consecutive th powers, and repeatedly take pairwise differences, you always end up with , … Continue reading

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PIE: proof by counting

Recall the setup: we have a universal set and a collection of subsets , , , and so on, up to . PIE claims that we can compute the number of elements of that are in none of the (that … Continue reading

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PIE: proof by algebra

In my previous post I stated a very formal, general form of the Principle of Inclusion-Exclusion, or PIE.1 In this post I am going to outline one proof of PIE. I’m not going to give a completely formal proof, because … Continue reading

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Formal PIE

I’ve been talking informally about the Principle of Inclusion-Exclusion but I realized it would be useful to state it more formally before proceeding to some proofs. The only problem is that a fully formal statement of PIE has a lot … Continue reading

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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

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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

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