SSRP Abstract

Board 19: Partial Sums in Reverse Lexicographic Order

Student Scientist: Rachel Leslie ’23
Research Mentor: Anna Pun (Department of Mathematics, Baruch College)

You can split any number into smaller numbers, such as 7 into 2+5 or 3+4. These are called partitions, and with certain equations, you can assign each partition a 1 or -1. Using a particular ordering, we’re wondering if when you add these 1’s and -1’s together consecutively, you get zero or bigger at every step.

Each partition of an integer can be given a sign, 1 or -1, using the integer n, the length of the partition l(λ), and the signum function sgn(σ)= (-1)^n-l(σ). Euler proved that the sum of all these signs is always a nonnegative number. Our conjecture states that using reverse lexicographic ordering and working from the smallest partitions to the largest, all partial sums of signs will be nonnegative as well. Unfortunately, this conjecture is still unproven. This research will help us understand the significance of reverse lexicographic ordering of partitions and why certain things are possible in this ordering but not others.