The Euler-Glaisher Theorem over Totally Real Number Fields

2023/11/30 by Jang, Se Wook, Kim, Byeong Moon, Kim, Kwang Hoon
#11P84 #11R80 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2311.18515

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=(λ₁,λ₂,…,λ_r) for some totally positive integers λ_i such that δ=λ₁+λ₂+⋯+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=x₁+2x₂+⋯+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=(λ₁,λ₂,…,λ_r) of δ such that λ_i+1-λ_i is totally positive or 0.

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