Sometimes, if you take the p-adic valutation of every number in a sequence, you get really pretty patterns. For example, if we graph the 3-adic valuation of the integers, we get this pretty pattern:

We can also look at a sequence of binomial coefficients. This picture shows the 3-adic valuation of the integers choose 7

Which has similar characteristics.

The Stirling numbers of the second kind have some similar characteristics, but are a little less predictable.

However, last summer, 3 REU students, Ana Berrizbeitia, Alexander Moll, and Laine Noble, researched p-adic valuation of Stirling numbers of the second kind, and were able to formalize the behavior.

My goal is to do something similar for the even less well behaved Eulerian Numbers.

These three types of sequences, Binomial Coefficients, Stirling numbers of the second kind, and Eulerian numbers are all examples of triangular recurrences. My REU project is just one case of a more general problem: Can the p-adic valuation of all triagular recurrences be neatly described? and if the answer is no, then what is it about these sequences that makes them well behaved?

I did an REU in 2006 about Matrix Polynomials with Robert Wilson. That project is here

Also, check out my skyscraper puzzles (Skyscraper Puzzles)