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<p>In this paper, we introduce alternative representations for the generating function of the number of cubic partitions of an integer. These new representations lead to novel formulas and provide a fresh combinatorial interpretation of cubic partitions as color partitions into distinct parts. We also obtain analogous results regarding the number of parts of size <i>d</i> colored identically in the cubic partitions of <i>n</i> and the number of cubic partitions of <i>n</i> that exclude parts of size <i>d</i> colored identically. Additionally, a new connection between an alternative sum of divisors and the 2-adic valuation is established. Furthermore, we present two open problems related to the positivity of truncated theta series within this framework.</p>

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Cubic partitions in terms of distinct partitions

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年4月3日

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  • Mircea Merca

摘要

In this paper, we introduce alternative representations for the generating function of the number of cubic partitions of an integer. These new representations lead to novel formulas and provide a fresh combinatorial interpretation of cubic partitions as color partitions into distinct parts. We also obtain analogous results regarding the number of parts of size d colored identically in the cubic partitions of n and the number of cubic partitions of n that exclude parts of size d colored identically. Additionally, a new connection between an alternative sum of divisors and the 2-adic valuation is established. Furthermore, we present two open problems related to the positivity of truncated theta series within this framework.