Rank functions and invariants of delta-matroids

TL;DR

This paper introduces a rank function framework for delta-matroids, explores their rank generating functions, and establishes a log-concavity property using Lorentzian polynomial theory.

Contribution

It provides a new axiomatization for delta-matroids' rank functions and links their rank generating functions to combinatorial invariants.

Findings

Axiomatization of delta-matroids' rank functions

Connection between rank generating functions and independent sets

Log-concavity of the evaluated rank generating function

Abstract

In this note, we give a rank function axiomatization for delta-matroids and study the corresponding rank generating function. We relate an evaluation of the rank generating function to the number of independent sets of the delta-matroid, and we prove a log-concavity result for that evaluation using the theory of Lorentzian polynomials.