Constructive Method for Finding the Coefficients of a Divided Symmetrization

TL;DR

This paper introduces a constructive method to determine the coefficients of a divided symmetrization operator for specific partitions and graphs, providing combinatorial tools for expansion over Schur functions.

Contribution

It presents a novel combinatorial approach for explicitly computing the Schur function expansion coefficients of divided symmetrizations for hook-shaped partitions with first part 2.

Findings

Explicit expansion formulas for specific partitions and graphs

A combinatorial construction for expansion terms

A computational method for coefficients

Abstract

We consider a type of divided symmetrization $\overrightarrow{D}_{\lambda,G}$ where $\lambda$ is a nonincreasing partition on $n$ and where $G$ is a graph. We discover that in the case where $\lambda$ is a hook shape partition with first part equal to 2, we may determine the expansion of $\overrightarrow{D}_{\lambda,G}$ over the basis of Schur functions. We show a combinatorial construction for finding the terms of the expansion and a second construction that allows computation of the coefficients.