A classical result in elementary number theory says that the product of the pairwise differences between any given $n + 1$ integers is divisible by the product of the pairwise differences between $0, 1, \ldots, n$. In the late 90s, Manjul Bhargava developed this much further into a theory of "generalized factorials", in particular giving a quasi-algorithm for finding the gcd of the products of the pairwise differences between any $n + 1$ integers in $S$, where $n$ is a given number and $S$ is a given set of integers. In this talk, I will explain why this is actually a combinatorial question in disguise, and how to answer it in full generality (joint work with Fedor Petrov). The general setting is a finite set $E$ equipped with weights (every element of $E$ has a weight) and distances (any two distinct elements of $E$ have a distance), where the distances satisfy the ultrametric triangle inequality.
The question is then to find a subset of $E$ of given size that has maximum perimeter (i.e., sum of weights of elements plus their pairwise distances). It turns out that all such subsets form a "strong greedoid" – a type of set system particularly adapted to optimization. Even better, this greedoid is a "Gaussian elimination greedoid" – which, roughly speaking, means that the problem reduces to linear algebra. If time allows, I will briefly discuss another closely related greedoid coming from a rather similar problem in phylogenetics. (This is mostly due to Manson, Moulton, Semple and Steel.)
Darij Grinberg