The Hopf monoid of Megagreedoids

TL;DR

This paper introduces megagreedoids, a new combinatorial structure generalizing several known objects, and demonstrates their Hopf monoid structure along with a quasisymmetric function invariant.

Contribution

It defines megagreedoids, establishes their Hopf monoid structure, and constructs a quasisymmetric function invariant with positive fundamental basis expansion.

Findings

Megagreedoids generalize polymatroids, megamatroids, and greedoids.

A quasisymmetric function invariant with positive expansion is constructed.

Megagreedoids form a Hopf monoid structure.

Abstract

We introduce megagreedoids, which generalize polymatroids, megamatroids, and greedoids. We define a quasisymmetric function invariant for a megagreedoid, and show that it has a positive expansion in the basis of fundamental quasisymmetric functions. Our proof involves lexicographic shellability. We also show that megagreedoids form a Hopf monoid. A running example is a megagreedoid associated to a rooted connected graph, and the resulting generalization of the chromatic symmetric function.