Category Archives: combinatorics

Möbius inversion

In my last post we saw that , that is, the Möbius function is the inverse of with respect to Dirichlet convolution. This directly leads to an interesting principle called Möbius inversion. Möbius inversion. Suppose is defined for as the … Continue reading

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

Dirichlet convolution and the Möbius function

Recall from last time that the Dirichlet convolution of two functions and is written and defined by: where the sum is taken over all possible factorizations of into a product of positive integers. Last time we saw that is commutative … Continue reading

Posted in combinatorics, proof | Tagged , , , | 1 Comment

Dirichlet convolution

Let and be two functions defined on the positive integers. Then the Dirichlet convolution of and , written , is another function on the positive integers, defined as follows: The sum is taken over all possible factorizations of into a … Continue reading

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

The Möbius function proof, part 2 (the subset parity lemma)

Continuing from my previous post, we are in the middle of proving that satisfies the same equation as , that is, and that therefore for all , that is, is the sum of all the th primitive roots of unity. … Continue reading

Posted in arithmetic, combinatorics, complex numbers, primes, proof | Tagged , , , , , , , , , | 2 Comments

The birthday candle problem: solution

Recall the birthday candle problem I wrote about in a previous post: A birthday cake has lit candles. At each step you pick a number uniformly at random and blow out candles. If any candles remain lit, the process repeats … Continue reading

Posted in combinatorics, probability, solutions | Tagged , , , | 5 Comments

The birthday candle problem

After a 1.5-month epic journey1, I am finally settling into my new position at Hendrix College. Here’s a fun problem I just heard from my new colleage Mark Goadrich: A birthday cake has lit candles. At each step you pick … Continue reading

Posted in challenges, combinatorics, probability | Tagged , , , | 5 Comments

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