Three Combinatorial Algorithms for the Cave Polynomial of a Polymatroid

TL;DR

This paper explores three combinatorial algorithms for computing the cave polynomial of a polymatroid, linking it to the Snapper polynomial and advancing understanding of polymatroid algebraic properties.

Contribution

It introduces three new combinatorial algorithms for the cave polynomial and connects these formulas to the Snapper polynomial, enhancing computational methods.

Findings

Three combinatorial algorithms for the cave polynomial

Interpretation of the Snapper polynomial via these formulas

Enhanced understanding of polymatroid algebraic structures

Abstract

The cave polynomial of a polymatroid was recently introduced and used to study the syzygies of polymatroidal ideals. We study the combinatorial relationships between three formulas for the cave polynomial. As an application, we interpret the Snapper polynomial in terms of these three formulas.