Asymptotic and exact results on the complexity of the Novelli-Pak-Stoyanovskii algorithm

TL;DR

This paper provides new asymptotic and exact complexity results for the Novelli-Pak-Stoyanovskii algorithm, a sorting method for Young tableaux, including average and worst case analyses as shapes grow.

Contribution

It introduces asymptotic complexity results for fixed shape limits and an exact formula for two-row shapes, addressing open questions in the field.

Findings

Asymptotic average and worst case complexity results derived.

Exact average case complexity formula for two-row shapes proved.

Answers to previously posed open questions by Krattenthaler and Müller.

Abstract

The Novelli-Pak-Stoyanovskii algorithm is a sorting algorithm for Young tableaux of a fixed shape that was originally devised to give a bijective proof of the hook-length formula. We obtain new asymptotic results on the average case and worst case complexity of this algorithm as the underlying shape tends to a fixed limit curve. Furthermore, using the summation package Sigma we prove an exact formula for the average case complexity when the underlying shape consists of only two rows. We thereby answer questions posed by Krattenthaler and M\"uller.