The Euler-Glaisher Theorem over Totally Real Number Fields

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 $δ$ over $K$ is $λ=(λ_1,λ_2,\ldots,λ_r)$ for some totally positive integers $λ_i$ such that $δ=λ_1+λ_2+\cdots+λ_r$. We find an identity to explain the number of partitions of $δ$ 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 $δ=x_1+2x_2+\cdots+nx_n$ with $x_i$ totally positive or $0$ is equal to that of chain partitions of $δ$. A chain partition of $δ$ is a partition $λ=(λ_1,λ_2,\ldots,λ_r)$ of $δ$ such that $λ_{i+1}-λ_i$ is totally positive or $0$.