Abstract

In this paper, we show that the difference between the number of parts in the odd partitions of n and the number of parts in the distinct partitions of n satisfies Euler’s recurrence relation for the partition function p(n) when n is odd. A decomposition of this difference in terms of the total number of parts in all the partitions of n is also derived. In this context, we conjecture that for k > 0, the series

has non-negative coefficients.

中文翻译：

---

关于划分为不同部分和奇数部分

摘要

在本文中，我们表明，部分中的奇数的分区的数目之间的二FF erence Ñ和部件的在的不同分区的数目Ñ满足欧拉为分区函数递归关系p（Ñ）时Ñ为奇数。还导出了根据n 的所有分区中的部分总数对这种差异进行的分解。在这种情况下，我们推测对于k > 0，级数

有非负系数ffi。