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<p>A cubic partition is an integer partition wherein the even parts can appear in two colors. Recently, Merca investigated the partition function <i>A</i>(<i>n</i>) which counts the difference between the number of cubic partitions of <i>n</i> into an even number of parts and the number of cubic partitions of <i>n</i> into an odd number of parts. At the end of his paper, he posed a conjecture on linear inequalities involving <i>A</i>(<i>n</i>). In this paper, we confirm Merca’s conjecture by using some results on truncated sums.</p>

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Proof of a conjecture of Merca on the cubic partition function

  • 期刊论文

  • 发表日期：

    2025年4月15日

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  • Jingzhao Zhang,
  • Olivia X. M. Yao

摘要

A cubic partition is an integer partition wherein the even parts can appear in two colors. Recently, Merca investigated the partition function A(n) which counts the difference between the number of cubic partitions of n into an even number of parts and the number of cubic partitions of n into an odd number of parts. At the end of his paper, he posed a conjecture on linear inequalities involving A(n). In this paper, we confirm Merca’s conjecture by using some results on truncated sums.