Seminar

# Gaussian elimination greedoids from ultrametric spaces: the combinatorics of Bhargava's generalized factorials

#### Algebraic and Enumerative Combinatorics

#### 10 March 11:00 - 11:50

#### Darij Grinberg - Drexel University

Consider a finite set $E$. Assume that each $e \in E$ has a "weight" $w(e) \in \mathbb{R}$ assigned to it, and any two distinct $e, f \in E$ have a "distance" $d(e,f) = d(f,e) \in \mathbb{R}$ assigned to them, such that the distances satisfy the ultrametric triangle inequality $d(a,b) \leq \max\{d(a,c), d(b,c)\}$. How would you find a subset of $E$ of given size with maximum perimeter (where the perimeter is defined by summing the weights of all elements and their pairwise distances)? It turns out that all such subsets form a greedoid -- an "order-aware" variant of a matroid; moreover, this greedoid is a Gaussian elimination greedoid (an analogue of matroid representability) over any sufficiently large field. The question has its origins in Manjul Bhargava's construction of generalized factorials, but is also a close relative of a problem in phylogenetics studied by Moulton, Semple and Steel. The talk will feature a quick introduction to greedoids; I will also propose an open question about strong greedoids in general.

Organizers

Sara Billey

University of Washington

Petter Brändén

KTH Royal Institute of Technology

Sylvie Corteel

Université Paris Diderot, Paris 7

Svante Linusson

KTH Royal Institute of Technology