Duality Law for Stirling Numbers
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This theorem requires a proof.
It is unclear exactly how the Stirling numbers are extended to the negative integers. Knuth's exposition is uncharacteristically non-explicit.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.
Theorem
For all integers $n, m \in \Z$:
- $\displaystyle \left\{ {n \atop m}\right\} = \left[{-m \atop -n}\right]$
where:
- $\displaystyle \left\{ {n \atop m}\right\}$ denotes a Stirling number of the second kind
- $\displaystyle \left[{n \atop m}\right]$ denotes an unsigned Stirling number of the first kind.
Proof
It is unclear exactly how the Stirling numbers are extended to the negative integers. Knuth's exposition is uncharacteristically non-explicit.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.
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Sources
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: $(58)$
