Category Archives: combinatorics

PIE day

[This is part six in an ongoing series; previous posts can be found here: Differences of powers of consecutive integers, Differences of powers of consecutive integers, part II, Combinatorial proofs, Making our equation count, How to explain the principle of … Continue reading

Posted in combinatorics, counting | Tagged , , , , | 2 Comments

The Steinhaus-Johnson-Trotter algorithm

In a previous post I posed the question: is there a way to list the permutations of in such a way that any two adjacent permutations are related by just a single swap of adjacent numbers? (Just for fun, let’s … Continue reading

Posted in combinatorics, pattern, solutions | Tagged , , , , , | 9 Comments

Permuting permutations

As you probably know, there are ( factorial) different ways to put the numbers from through (or any set of distinct objects) in a list. For example, here are the different lists containing the numbers through : Each such list … Continue reading

Posted in challenges, combinatorics | Tagged , | 11 Comments

How to explain the principle of inclusion-exclusion?

I’ve been remiss in finishing my series of posts on a combinatorial proof. I still intend to, but I must confess that part of the reason I haven’t written for a while is that I’m sort of stuck. The next … Continue reading

Posted in combinatorics, meta | Tagged , , , | 7 Comments

Making our equation count

[This is post #4 in a series; previous posts can be found here: Differences of powers of consecutive integers, Differences of powers of consecutive integers, part II, Combinatorial proofs.] We’re still trying to find a proof of the equation which … Continue reading

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

Combinatorial proofs

Continuing from a previous post, we found that if we begin with th powers of consecutive integers and then repeatedly take successive differences, it seems like we always end up with the factorial of , that is, . We then … Continue reading

Posted in combinatorics, pictures, proof | Tagged , , | 12 Comments