Symmetry properties of the Novelli-Pak-Stoyanovskii algorithm

TL;DR

This paper explores the symmetry properties of the Novelli-Pak-Stoyanovskii algorithm, a bijective sorting method for standard Young tableaux, extending previous results with new bijective proofs.

Contribution

It extends prior work by providing new bijective proofs of the symmetry properties of the algorithm, deepening understanding of its combinatorial structure.

Findings

The algorithm exhibits notable symmetry properties.

New bijective proofs confirm previous symmetry results.

The complexity measure relates to the algorithm's exchange operations.

Abstract

The number of standard Young tableaux of a fixed shape is famously given by the hook-length formula due to Frame, Robinson and Thrall. A bijective proof of Novelli, Pak and Stoyanovskii relies on a sorting algorithm akin to jeu-de-taquin which transforms an arbitrary filling of a partition into a standard Young tableau by exchanging adjacent entries. Recently, Krattenthaler and M\"uller defined the complexity of this algorithm as the average number of performed exchanges, and Neumann and the author proved it fulfils some nice symmetry properties. In this paper we recall and extend the previous results and provide new bijective proofs.