Labelled trees and pairs of input-output permutations in priority queues

Abstract

A priority queue can transform a permutation π of a set of size n to some but not necessarily all permutations σ. A recent result of Atkinson and Thiyagarajah [1] states that the number of distinct pairs (π, σ) is (n+1)n−1. Recall that Cayley's Theorem ([2]) states that the number of labelled trees on n+1 nodes is also equal to (n+1)n−1. We present a direct correspondence between these labelled trees and these pairs of permutations and discuss related problems.