论文标题
在分区分为不同的部分和奇数部分
On the partitions into distinct parts and odd parts
论文作者
论文摘要
在本文中，我们表明，$ n $的奇数分区中的零件数量与$ n $不同分区中的零件数量之间的差异满足了$ n $奇怪的分区功能$ p（n）$的欧拉的复发关系。也得出了所有$ n $分区中零件总数的分解。在这种情况下，我们猜想，对于$ k> 0 $，该系列 $$ （q^2; q^2）_ \ infty \ sum_ {n = k}^\ infty \ frac {q^{{k \ select 2}+（k+1）n}}} {（q; q; q; q; q）_n} \ begin {bmatrix} n-1 \\ k-1 \ end {bmatrix} $$具有非负系数。
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 $$ (q^2;q^2)_\infty \sum_{n=k}^\infty \frac{q^{{k\choose 2}+(k+1)n}}{(q;q)_n} \begin{bmatrix} n-1\\k-1 \end{bmatrix} $$ has non-negative coefficients.
