The Euler-Glaisher Theorem over Totally Real Number Fields

TL;DR

This paper extends the classical Euler-Glaisher partition theorem to totally real number fields, establishing new identities and connections between partitions, ideals, and chain partitions in this algebraic setting.

Contribution

It generalizes the Euler-Glaisher Theorem to totally real number fields and links solutions of certain equations to chain partitions in this context.

Findings

Derived an identity for partitions avoiding a given ideal.

Generalized Euler-Glaisher Theorem over totally real fields.

Established equivalence between solutions to a sum equation and chain partitions.

Abstract

In this paper, we study the partition theory over totally real number fields. Let $K$ be a totally real number field. A partition of a totally positive algebraic integer $\delta$ over $K$ is $\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_r)$ for some totally positive integers $\lambda_i$ such that $\delta=\lambda_1+\lambda_2+\cdots+\lambda_r$. We find an identity to explain the number of partitions of $\delta$ whose parts do not belong to a given ideal $\mathfrak a$. We obtain a generalization of the Euler-Glaisher Theorem over totally real number fields as a corollary. We also prove that the number of solutions to the equation $\delta=x_1+2x_2+\cdots+nx_n$ with $x_i$ totally positive or $0$ is equal to that of chain partitions of $\delta$. A chain partition of $\delta$ is a partition $\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_r)$ of $\delta$ such that $\lambda_{i+1}-\lambda_i$ is totally positive or $0$.