Gr\"obner bases for staged trees

TL;DR

This paper studies the algebraic structure of staged trees, showing that balanced and stratified ones have toric ideals generated by quadratic Gr"obner bases with squarefree initial terms, using toric fiber product techniques.

Contribution

It establishes a quadratic Gr"obner basis generation result for toric ideals of balanced, stratified staged trees, advancing algebraic understanding of these combinatorial objects.

Findings

Toric ideals of balanced, stratified staged trees are generated by quadratic Gr"obner bases.

Initial terms of these bases are squarefree.

Proof utilizes Sullivant's toric fiber product construction.

Abstract

In this article we consider the problem of finding generators of the toric ideal associated to a combinatorial object called a staged tree. Our main theorem states that toric ideals of staged trees that are balanced and stratified are generated by a quadratic Gr\"obner basis whose intial terms are squarefree. The proof of this result is based on Sullivant's toric fiber product construction.