On the Independent Set Numbers of a Finite Matroid

Abstract

This chapter discusses the independent set numbers of a finite matroid. In a finite matroid of rank r , W k and I k denote the number of closed sets and the number of independent sets of rank k , where 0 ≤ k ≤ r. There are a number of interesting conjectures about these sequences. Rota conjectured that the sequence ( W k ) of Whitney numbers is unimodal, and there is evidence to support the stronger conjecture that it is logarithmically concave. Another conjecture is that W k ≤ W r – k when k ≤ r – k . Analogous conjectures have been made for the sequence ( I k ) of independent set numbers. The chapter discusses different type of strengthening of logarithmic concavity conjecture of Mason.