# A note on the signature representations of the symmetric groups

## TL;DR

This paper establishes a precise criterion for when a composition derived from a partition can have all partial sums non-divisible by a prime p, contributing to the understanding of symmetric group representations.

## Contribution

It provides a necessary and sufficient condition linking partitions and compositions with divisibility constraints, advancing the theory of symmetric group signatures.

## Key findings

- Characterizes when compositions from partitions avoid divisibility by p
- Provides a complete criterion for the existence of such compositions
- Enhances understanding of symmetric group signature representations

## Abstract

For a partition {\lambda} and a prime p, we prove a necessary and sufficient condition for there exists a composition {\delta} such that {\delta} can be obtained from {\lambda} after rearrangement and all the partial sums of {\delta} are not divisible by p.