The ordered Bell numbers as weighted sums of odd or even Stirling numbers of the second kind

TL;DR

This paper proves identities linking ordered Bell numbers to weighted sums of Stirling numbers of the second kind, revealing new combinatorial relationships and formulas.

Contribution

It introduces novel identities connecting ordered Bell numbers with sums of Stirling numbers, expanding understanding of their combinatorial structure.

Findings

Established the identity $ extstyle\sum_{k=1}^{n/2} S(n,2k)(2k-1)! = B(n-1)$

Derived an analogous identity for sums over odd $k$'s

Enhanced the combinatorial interpretation of ordered Bell numbers

Abstract

For the Stirling numbers of the second kind $S(n,k)$ and the ordered Bell numbers $B(n)$, we prove the identity $\sum_{k=1}^{n/2} S(n,2k)(2k-1)! = B(n-1)$. An analogous identity holds for the sum over odd $k$'s.