Many combinatorial functions related to integer partitions have a surprising kind of regularity in their divisibility properties. For example, the Ramanujan Congruences say that p(5n+4) is a multiple of 5 for all n, where p(n) is the partition counting function. A combinatorial explanation of this fact is given by Dyson’s rank function, which splits the set of partitions of 5n+4 into 5 sets of equal size. We will use computational methods to explore some of the arithmetic patterns related to ranks of partitions.
Students should have some experience with discrete math via 213 or 347 or something similar, either previously or concurrently. Experience with modular arithmetic and/or combinatorics would be particularly helpful.
We will use Mathematica throughout the semester. Experience programming would be very helpful.