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Is there a standard compact notation for the numbers $$ ?(n,k)=\sum_{i=0}^k(-1)^i\binom ki(k-i)^n\equiv k!{n\brace k} $$ where ${n\brace k}$ is the Stirling number of the second kind? In probabilistic applications the combination (which counts the number of ways to partition a set of $n$ objects into $k$ non-empty distinguishable subsets) seems to appear even more often than the Stirling numbers themselves. And if $k$ is a very long expression I would prefer to avoid to write it once more as $(\dots)!$

RobPratt
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user
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    Nothing standard, but I write $\operatorname{sur}\left(n,k\right)$ (as it counts surjective maps). – darij grinberg Mar 04 '21 at 11:10
  • https://math.stackexchange.com/questions/3250397/summing-s-n-m-sum-k-1n-1k-km-n-choose-k-for-mn-n – Z Ahmed Mar 04 '21 at 12:03
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    @ZAhmed I do not need to compute the value, I am looking for a compact notation for it. – user Mar 04 '21 at 12:04
  • @DanielV Thank you for advice. I have found a 1992 paper of Knuth where he at length discusses the Stirling numbers but never mentions their product with respective factorial, which combinatorial meaning is the distribution of objects in distinguishable boxes. – user Mar 04 '21 at 17:34
  • I think I might be remembering something different by knuth. – DanielV Mar 04 '21 at 17:38

1 Answers1

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The numbers you are using are the triangular arrray OEIS sequence A019538. As mentioned in an OEIS comment, they were known as the "differences of zero" denoted by $\,\Delta^k0^n.$ They are the numbers in the leading diagonal of the forward difference table of $\,x^n\,$ at $\,x=0.\,$

For example, when $\,n=4\!:$ $$ \Delta^k x^4|_{x=0} =: \Delta^k 0^4 = 0,\;1,\;14,\;36,\;24 $$ from the forward difference table: $$ \begin{matrix} \Delta^4x^4 &| &&&&& 24 \\ \Delta^3x^4 &| &&&& 36 && 60 \\ \Delta^2x^4 &| &&& 14 && 50 && 110 \\ \Delta^1x^4 &| && 1 && 15 && 65 && 175 \\ x^4 &| & 0 && 1 && 16 && 81 && 256 \\ x &| & 0 && 1 && 2 && 3 && 4 \end{matrix} $$

This notation is used in older books such as George Boole's A Treatise on The Calculus of Finite Differences. The Wikipedia article Finite difference explains some of this.

In the matter of how standard, compact or convenient this notation is in practice, there may be some differences of opinion.

Somos
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  • This is a very interesting notation. Never heard about this. But in practical applications it is not convenient as $n$ and $k$ may be rather complicated expressions. – user Mar 04 '21 at 12:56