On random shifted standard Young tableaux and 132-avoiding sorting networks

TL;DR

This paper investigates the asymptotic shape of random shifted standard Young tableaux of staircase shape and explores their connection to 132-avoiding sorting networks, revealing new limit shapes and adjacency properties.

Contribution

It establishes the limiting surface of shifted SYT of staircase shape and links these results to properties of 132-avoiding sorting networks, including limit shapes and adjacency counts.

Findings

Determined the limit shape of random shifted SYT of staircase shape.

Established limit shapes for trajectories and permutations in 132-avoiding sorting networks.

Showed that each row and column of such SYT contains exactly one adjacency on average.

Abstract

We study shifted standard Young tableaux (SYT). The limiting surface of uniformly random shifted SYT of staircase shape is determined, with the integers in the SYT as heights. This implies via properties of the Edelman-Greene bijection results about random 132-avoiding sorting networks, including limit shapes for trajectories and intermediate permutations. Moreover, the expected number of adjacencies in SYT is considered. It is shown that on average each row and each column of a shifted SYT of staircase shape contains precisely one adjacency.