Priority Queues and Permutations

Abstract

A priority queue transforms an input permutation $\sigma $ of some set of size n into an output permutation $\tau $ The set $R_n $ of such related pairs $(\sigma ,\tau )$ is studied. Efficient algorithms for determining $s(\tau ) = |\sigma :(\sigma ,\tau ) \in R_n |$, and $t(\sigma ) = |\tau :(\sigma ,\tau ) \in R_n |$ are given, a new proof that $|R_n | = (n + 1)^{n - 1} $ is given, and the transitive closure of $R_n $ is found.