Asymptotic expansions for factorial moments of some distributions in the analysis of algorithms

TL;DR

This paper derives asymptotic expansions for factorial moments of key distributions related to permutations and quicksort, using advanced singularity analysis of generating functions to deepen understanding of their probabilistic behavior.

Contribution

It introduces a method to obtain asymptotic factorial moments for permutation and sorting distributions using singularity analysis, extending previous approaches.

Findings

Asymptotic expansions for factorial moments of permutation cycle counts

Asymptotic expansions for inversion counts in permutations

Asymptotic expansions for comparisons in randomized quicksort

Abstract

We establish asymptotic expansions for factorial moments of following distributions: number of cycles in a random permutation, number of inversions in a random permutation, and number of comparisons used by the randomized quick sort algorithm.To achieve this we use singularity analysis of certain type of generating functions due to Flajolet and Odlyzko.