# Duality Law for Stirling Numbers

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## 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

## Sources

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