A Plancherel measure associated to set partitions and its limit

TL;DR

This paper introduces a superplancherel measure linked to set partitions within supercharacter theories, establishing a limit shape result and analyzing asymptotic behavior of related statistics.

Contribution

It defines a new superplancherel measure for set partitions and proves a limit shape theorem, extending understanding of supercharacter theory asymptotics.

Findings

Established a limit shape for random set partitions under the superplancherel measure.

Described asymptotic behavior of set partition statistics related to supercharacters.

Connected the new measure to prior uniform distribution results by other researchers.

Abstract

In recent years increasing attention has been paid on the area of supercharacter theories, especially to those of the upper unitriangular group. A particular supercharacter theory, in which supercharacters are indexed by set partitions, has several interesting properties, which make it object of further study. We define a natural generalization of the Plancherel measure, called superplancherel measure, and prove a limit shape result for a random set partition according to this distribution. We also give a description of the asymptotical behavior of two set partition statistics related to the supercharacters. The study of these statistics when the set partitions are uniformly distributed has been done by Chern, Diaconis, Kane and Rhoades.