Abstract

In this paper, we consider extensions of Spivey’s Bell number formula wherein the argument of the polynomial factor is translated by an arbitrary amount. This idea is applied more generally to the r-Whitney numbers of the second kind, denoted by W (n, k), where some new identities are found by means of algebraic and combinatorial arguments. The former makes use of infinite series manipulations and Dobinski-like formulas satisfied by W (n, k), whereas the latter considers distributions of certain statistics on the underlying enumerated class of set partitions. Further-more, these two approaches provide new ways in which to deduce the Spivey formula for W (n, k). Finally, we establish an analogous result involving the r-Lah numbers wherein the order matters in which the elements are written within the blocks of the aforementioned set partitions.

摘要

在本文中，我们考虑了 Spivey 贝尔数公式的扩展，其中多项式因子的参数被任意平移。这个想法更普遍地应用于第二类r -Whitney 数，用W ( n, k ) 表示，其中一些新恒等式是通过代数和组合论证找到的。前者利用W ( n, k ) 满足的无限级数操作和类似 Dobinski 的公式，而后者考虑某些统计数据在集合分区的基础枚举类上的分布。此外，这两种方法提供了推导出W ( n, k的 Spivey 公式的新方法）。最后，我们建立了一个涉及r -Lah 数的类似结果，其中在上述集合分区的块中写入元素的顺序很重要。