# Tiered trees, weights, and q-Eulerian numbers

## TL;DR

This paper introduces tiered trees as a generalization of maxmin trees, explores their combinatorial properties, and develops a new q-analogue of Eulerian numbers through weight functions.

## Contribution

It generalizes maxmin trees to tiered trees, establishes bijections with other combinatorial objects, and defines a novel q-analogue of Eulerian numbers.

## Key findings

- Weight 0 tiered trees biject with permutations
- New q-analogue of Eulerian numbers derived
- Connections to quiver representations and homogeneous varieties

## Abstract

Maxmin trees are labeled trees with the property that each vertex is either a local maximum or a local minimum. Such trees were originally introduced by Postnikov, who gave a formula to count them and different combinatorial interpretations for their number. In this paper we generalize this construction and define tiered trees by allowing more than two classes of vertices. Tiered trees arise naturally when counting the absolutely indecomposable representations of certain quivers, and also when one enumerates torus orbits on certain homogeneous varieties. We define a notion of weight for tiered trees and prove bijections between various weight 0 tiered trees and other combinatorial objects; in particular order n weight 0 maxmin trees are naturally in bijection with permutations on n-1 letters. We conclude by using our weight function to define a new q-analogue of the Eulerian numbers.