# $k$-partial permutations and the center of the wreath product $\mathcal{S}_k\wr \mathcal{S}_n$ algebra

## TL;DR

This paper introduces $k$-partial permutations to analyze the center of wreath product algebras, proving structure coefficients are polynomials with non-negative coefficients and connecting to shifted symmetric functions.

## Contribution

It generalizes partial permutations to $k$-partial permutations and establishes a polynomiality result for structure coefficients in wreath product centers.

## Key findings

- Structure coefficients are polynomials in $n$ with non-negative integer coefficients.
- The algebra $ ext{I}_ ext{infty}^k$ is isomorphic to shifted symmetric functions.
- Provides a universal algebra framework for the center of wreath product algebras.

## Abstract

We generalize the concept of partial permutations of Ivanov and Kerov and introduce $k$-partial permutations. This allows us to show that the structure coefficients of the center of the wreath product $\mathcal{S}_k\wr \mathcal{S}_n$ algebra are polynomials in $n$ with non-negative integer coefficients. We use a universal algebra $\mathcal{I}_\infty^k$ which projects on the center $Z(\mathbb{C}[\mathcal{S}_k\wr \mathcal{S}_n])$ for each $n.$ We show that $\mathcal{I}_\infty^k$ is isomorphic to the algebra of shifted symmetric functions on many alphabets.