The Stanely-F\'eray-\'Sniady formula for the generalized characters of the Symmetric Group

TL;DR

This paper extends the Stanley-Féray-Šniady formula for symmetric group characters to generalized characters, specifically spherical functions of a certain Gel'fand pair, providing a broader understanding of these mathematical objects.

Contribution

The paper introduces a natural extension of the Stanley-Féray-Šniady formula to generalized characters associated with a specific Gel'fand pair, expanding the theoretical framework.

Findings

Extended the formula to generalized characters

Connected the formula to spherical functions of a Gel'fand pair

Provided explicit expressions for these generalized characters

Abstract

We show that the explicit formula of Stanley-F\'eray-\'Sniady for the characters of the symmetric group have a natural extension to the generalized characters. These are the spherical functions of the unbalanced Gel'fand pair $(S_n\times S_{n-1},\text{diag}S_{n-1})$.