Identifying Self-Conjugate Partitions

TL;DR

This paper introduces a new method to identify self-conjugate partitions algebraically, avoiding the traditional visual approach using Young diagrams, by leveraging properties of symmetric shapes and multiplicities.

Contribution

It presents a novel algebraic criterion for determining self-conjugate partitions based on part multiplicities, bypassing the need for Young diagram visualization.

Findings

Established preliminary lemmas and theorems on symmetric shapes.

Proved that adding multiplicities of parts can identify self-conjugacy.

Provided an algebraic method for verification of self-conjugate partitions.

Abstract

A partition of a positive integer $n$ is defined as a non-increasing sequence $P = [y_0, y_1, ..., y_m]$ of positive integers which sum to $n$, where the $y_i$ are called the $parts$ of the partition. A Young diagram is a visual representation of a partition using rows of boxes, where each row of boxes corresponds to a part. The conjugate partition is similar to a transpose of a matrix; we switch the rows with columns, or the index of a part with the part itself. Self-conjugate partitions are partitions that are equal to their conjugate; previously, the only known way to verify whether a partition is self-conjugate was through the use of a Young diagram. In this research, by proving preliminary lemmas and theorems about easily identifiable shapes which are symmetric, we come to the main result: by simply adding the multiplicities of parts appropriately, we can show whether or not a partition is self-conjugate without the use of a Young diagram.