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Category Archives: combinatorics
A few words about PWW #33: subset permutations
My previous post showed four rows of diagrams, where the th row (counting from zero) has diagrams with dots. The diagrams in the th row depict all possible paths that start at the top left dot, end at the top … Continue reading
A combinatorial proof: PIE a la mode!
Continuing from my last post in this series, we’re trying to show that , where is defined as which is what we get when we start with a sequence of consecutive th powers and repeatedly take successive differences. Recall that … Continue reading
Posted in arithmetic, combinatorics, proof
Tagged consecutive, difference, function, integers, matching, powers
Comments Off on A combinatorial proof: PIE a la mode!
A combinatorial proof: counting bad functions
In a previous post we derived the following expression: . We are trying to show that , in order to show that starting with a sequence of consecutive th powers and repeatedly taking successive differences will always result in . … Continue reading
Posted in arithmetic, combinatorics, proof
Tagged consecutive, difference, function, integers, matching, powers
1 Comment
A combinatorial proof: functions and matchings
We’re trying to prove the following equality (see my previous post for a recap of the story so far): In particular we’re trying to show that the two sides of this equation correspond to two different ways to count the … Continue reading
Posted in arithmetic, combinatorics, proof
Tagged consecutive, difference, function, integers, matching, powers
5 Comments
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
Posted in arithmetic, combinatorics, proof
Tagged consecutive, difference, integers, powers
1 Comment
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
Posted in arithmetic, combinatorics, proof
Tagged consecutive, difference, integers, powers
11 Comments
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
PIE: proof by algebra
In my previous post I stated a very formal, general form of the Principle of InclusionExclusion, 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
Formal PIE
I’ve been talking informally about the Principle of InclusionExclusion 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
Posted in combinatorics, pattern
Tagged equation, exclusion, formal, inclusion, PIE, sets
5 Comments