Tiered trees and Theta operators

TL;DR

This paper uses Theta operators to derive a symmetric function formula for counting tiered trees, connecting combinatorial structures with algebraic operators and proposing a broader conjecture related to the Delta conjecture.

Contribution

It introduces a new symmetric function formula for tiered trees using Theta operators and formulates a general conjecture extending this enumeration.

Findings

Derived a symmetric function formula for tiered trees.

Formulated a conjecture extending the enumeration result.

Potential insights into the Delta conjecture.

Abstract

In [Dugan-Glennon-Gunnells-Steingrimsson-2019], the authors introduce tiered trees to define combinatorial objects counting absolutely indecomposable representations of certain quivers, and torus orbits on certain homogeneous varieties. In this paper, we use Theta operators, introduced in [D'Adderio-Iraci-VandenWyngaerd-Theta-2021], to give a symmetric function formula that enumerates these trees. We then formulate a general conjecture that extends this result, a special case of which might give some insight about how to formulate a unified Delta conjecture [Haglund-Remmel-Wilson-2018].