Tiered tree, Parking function and Postnikov-Shapiro algebra

TL;DR

This paper explores the combinatorial structures of tiered trees, parallelogram polyominoes, and parking functions, establishing bijections and algebraic descriptions to advance understanding of their interrelations.

Contribution

It introduces a bijection between parallelogram polyominoes and graphical parking functions and characterizes tiered graphs using Whitney's operations.

Findings

Established a bijection between parallelogram polyominoes and parking functions.

Defined the space mathcal{S}_G for complete tiered graphs.

Described tiered graphs in terms of Whitney's operations.

Abstract

Tiered trees were introduced as a combinatorial object for counting absolutely indecomposable representation of certain quivers and torus orbit of certain homogeneous variety. In this paper, we define a bijection between the set of parallelogram polyominoes and graphical parking functions. Moreover, we defined the space $\mathcal{S}_{G}$ for complete tiered graphs and described tiered graphs in terms of Whitney's operations.