# Toward the Schur expansion of Macdonald polynomials

## TL;DR

This paper provides a combinatorial formula for expanding Macdonald polynomials into Schur functions for specific partitions, unifying cases and confirming a conjecture about dual equivalence classes.

## Contribution

It introduces an explicit combinatorial formula for the Schur expansion of Macdonald polynomials for partitions with second part at most two, unifying hook and two-column cases.

## Key findings

- Derived a uniform Schur expansion formula for specific Macdonald polynomials.
- Proved that generalized dual equivalence classes are unions of standard classes in certain cases.
- Confirmed an earlier conjecture regarding dual equivalence classes.

## Abstract

We give an explicit combinatorial formula for the Schur expansion of Macdonald polynomials indexed by partitions with second part at most two. This gives a uniform formula for both hook and two column partitions. The proof comes as a corollary to the result that generalized dual equivalence classes of permutations are unions of standard dual equivalence classes of permutations for certain cases, establishing an earlier conjecture of the author.