The Euler-Glaisher Theorem over Totally Real Number Fields

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 $\lambda =({\lambda }_1,{\lambda }_2,\dots ,{\lambda }_r)$ 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 $\delta =x_1+2x_2+\cdots +n{x}_n$ with $x_i$ totally positive or $0$ is equal to that of chain partitions of $\delta$. A chain partition of $\delta$ is a partition $\lambda =({\lambda }_1,{\lambda }_2,\dots ,{\lambda }_r)$ of $\delta$ such that ${\lambda }_{i+1}-{\lambda }_i$ is totally positive or $0$.