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Tag Archives: binomial coefficients
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 binomial coefficients, combinatorics, functions, matching, permutation
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 binomial coefficients, combinatorial proof, identity
12 Comments
Differences of powers of consecutive integers, part II
If you spent some time playing around with the procedure from Differences of powers of consecutive integers (namely, raise consecutive integers to the th power, and repeatedly take pairwise differences until reaching a single number) you probably noticed the curious … Continue reading
Posted in arithmetic, iteration, pascal's triangle
Tagged binomial coefficients, consecutive, difference, integers, powers
3 Comments
Some words about Post without words #2
My previous post displayed this picture: As Yuriy Kashnikov guessed, I made this picture using diagrams, a Haskell library I am developing for creating images like this. (You can see the source code for this picture here.) If you haven’t … Continue reading
Posted in counting, pascal's triangle, pattern, pictures
Tagged binomial coefficients, diagram, Hasse, Pascal, subset
4 Comments
Distributing cookies: solutions
And now for some solutions to the cookie distribution problem. I’m actually going to describe four different methods of solution, and thereby (re)discover some nice combinatorial identities along the way. This is what I love about combinatorics—you discover all this … Continue reading
Posted in counting, proof, solutions
Tagged binomial coefficients, combinatorial identities, combinatorics, cookies, counting, distribution
4 Comments