论文标题
通过扁平分区进行分配
Run Distribution Over Flattened Partitions
论文作者
论文摘要
对扁平分区的研究是当前研究的活跃领域。在本文中，我们的研究意外导致我们进入OEI数字A124324。我们提供了这些数字的新组合解释。在单独的部分中也给出了超过$ [n+1] $的扁平分区与$ [n] $的分区之间的组合培训。我们介绍了$ f_ {n，k} $的数字，该数字计算了具有$ k $运行的$ [n] $上面的平坦分区的数量。我们给出了定义它们的复发关系，以及它们以差异形式的指数生成函数。如果建立了其封闭形式，则应对此表示赞赏。我们将结果扩展到平坦的分区，其中第一个$ S $整数属于不同的运行。给出了组合证明。
The study of flattened partitions is an active area of current research. In this paper, our study unexpectedly leads us to the OEIS numbers A124324. We provide a new combinatorial interpretation of these numbers. A combinatorial bijection between flattened partitions over $[n+1]$ and the partitions of $[n]$ is also given in a separate section. We introduce the numbers $f_{n, k}$ which count the number of flattened partitions over $[n]$ having $k$ runs. We give recurrence relations defining them, as well as their exponential generating function in differential form. It should be appreciated if its closed form is established. We extend the results to flattened partitions where the first $s$ integers belong to different runs. Combinatorial proofs are given.
