Alex Schaefer (Binghamton)

Crapo's Beta Invariant for Matroids

Abstract for the Combinatorics Seminar
2013 November 19

A matroid is an abstraction of linear independence of a finite set of vectors. There is a numerical invariant β of a matroid that has many useful implications. For instance, it tells whether the matroid splits into smaller matroids. After setup and a few results (regarding items such as partial differentiation on a rank function and factorization of matroids), I will define β, prove several nice properties, and do some calculations.

The talk will continue into a second half, where I will prove some of the deeper properties of β such as its non-negativity and duality invariance. I will also discuss connections between β and the more familiar Tutte polynomial and rank generating polynomial, and give some more complex computations regarding β for several different classes of matroids.

This is Mr. Schaefer's examination for Admission to Candidacy. The examining committee is Laura Anderson, Vaidy Sivaraman, and Thomas Zaslavsky (chair).


To the Combinatorics Seminar Web page.