TL;DR
The paper introduces the priority lattice, a new graded lattice structure related to parking functions, and analyzes its properties including M"obius function and characteristic polynomials.
Contribution
It establishes a bijection between maximal chains of the priority lattice and parking functions, revealing new combinatorial connections.
Findings
Maximal chains are labeled by parking functions.
The lattice is graded and its M"obius function is computed.
Principal ideals' chains are labeled by partial parking functions.
Abstract
We introduce the priority lattice, a structure arising from the priority search algorithm on rooted trees and forests. We prove bijectively that its maximal chains are labeled by parking functions, and that the maximal chains of its principal ideals are labeled by partial parking functions. We establish that it is a graded lattice and compute its M\"obius function and characteristic polynomials.
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