Duality Law for Stirling Numbers

Theorem

For all integers $n, m \in \Z$:

$\ds {n \brace m} = {-m \brack -n}$

where:

$\ds {n \brace m}$ denotes a Stirling number of the second kind

$\ds {n \brack m}$ denotes an unsigned Stirling number of the first kind.

Proof

This theorem requires a proof. In particular: 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. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: $(58)$