The ordered Bell numbers as weighted sums of odd or even Stirling numbers of the second kind
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.
- Publication:
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arXiv e-prints
- Pub Date:
- September 2021
- DOI:
- 10.48550/arXiv.2109.12705
- arXiv:
- arXiv:2109.12705
- Bibcode:
- 2021arXiv210912705S
- Keywords:
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- Mathematics - Combinatorics
- E-Print:
- 3 pages