The Priority Lattice

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 Möbius function and characteristic polynomials.

Figures (10)

• Figure 1: An ordered forest and its priority forest.
• Figure 2: The Hasse diagram of $\Pi(2)$, its partial edge labeling, and its Möbius function values.
• Figure 3: The Hasse diagram of $\Pi(3)$, its partial edge labeling, and its Möbius function values.
• Figure 4: An interval $[\hat{0}, T]$, its maximal chains and corresponding rooted trees, and some additional examples at the forest level.
• Figure 5: In the central column, a parking function as a sequence of partial parking functions. In the left column, the corresponding sequence of priority forests. In the right column, the corresponding sequence of ordered forests under the weary bijection.

Theorems & Definitions (49)

• Example 2.1
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Theorem 3.4
• proof
• Example 4.1