Hello everyone!

This week, I'd like to talk about permutations.  A permutation of a list is another list with the same elements in some order.  For instance, imagine a deck of playing cards.  Any way you shuffle the cards, you get the same cards, but you might get a different order.  So the resulting ordering is a permutation of the original ordering.

It turns out that for permutations, the specific items in our list don't matter all that much.  We can rename the items in the list however we like.  For instance, we could take our deck of cards and give each a different color, or we could number them .  Then we can think of shuffling the cards as rearranging colors or permuting the numbers through .

Continue reading Permutations →

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Hello everyone!

Today, I'd like to talk about one of the most fundamental concepts in combinatorics: binomial coefficients.  We define (read " choose ") to be the number of ways to choose of things.  So, for example, there are ways to choose of blocks to color red, as shown below.

Continue reading Choose Your Own Mathventure →

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Hello, everyone!

As I mentioned in my opening post, one of my interests is combinatorics.  Basically, it amounts to counting things cleverly.  It's probably not the most practical field (although it has its uses in statistics and quantum physics), but I find that it has some of the most beautiful and intuitive results.

One of the most studied topics in combinatorics is the Fibonacci sequence.  We start with then we get each other term by adding the previous two.  So the first few terms are .  More algebraically, we have and   Those are nice numbers, I guess, but what do they mean? Continue reading Fibonacci Tilings →

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