Definition:Stirling Numbers of the Second Kind/Definition 2
Definition
Stirling numbers of the second kind are defined as the coefficients $\ds {n \brace k}$ which satisfy the equation:
- $\ds x^n = \sum_k {n \brace k} x^{\underline k}$
where $x^{\underline k}$ denotes the $k$th falling factorial of $x$.
Notation
The notation $\ds {n \brace k}$ for Stirling numbers of the second kind is that proposed by Jovan Karamata and publicised by Donald E. Knuth.
Other notations exist.
Usage is inconsistent in the literature.
Also see
Source of Name
This entry was named for James Stirling.
Historical Note
This formula for the Stirling numbers of the second kind:
- $\ds x^n = \sum_k {n \brace k} x^{\underline k}$
was the reason James Stirling started his studies of the Stirling numbers in the first place.
They were studied in detail in his Methodus Differentialis of $1730$.
Technical Note
The $\LaTeX$ code for \(\ds {n \brace k}\) is \ds {n \brace k}
.
The braces around the n \brace k
are important.
The \ds
is needed to create the symbol in its proper house display style.
Sources
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: $(45)$