# Stirling numbers of the second kind

This online calculator outputs Stirling numbers of the second kind for the given n

### This page exists due to the efforts of the following people:

#### Timur

Created: 2019-12-22 14:55:49, Last updated: 2021-03-06 09:03:41

This content is licensed under Creative Commons Attribution/Share-Alike License 3.0 (Unported). That means you may freely redistribute or modify this content under the same license conditions and must attribute the original author by placing a hyperlink from your site to this work https://planetcalc.com/8529/. Also, please do not modify any references to the original work (if any) contained in this content.

In combinatorics, a Stirling number of the second kind (or Stirling partition number) is the number of ways to partition a set of n objects into k non-empty subsets and is denoted by S(n,k)1. This online calculator calculates the Stirling number of the second kind for the given n, for each k from 0 to n and outputs results into a table. Note that this calculator uses the "big integers" library (see Tips and tricks #9: Big numbers), so you can try pretty big n values.

For example, the number of ways to partition a set of 100 objects into 28 non-empty subsets is 77697 3005359874 5155212806 6127875847 8739787812 8370115840 9749925701 0238608628 9805848025 0748224048 4354517896 0761551674. A combinatorial explosion, that is :)

For those who curious, the explicit formula is listed below the calculator.

#### Stirling numbers of the second kind

The file is very large. Browser slowdown may occur during loading and creation.

### Stirling numbers formula

$\left\{{n \atop k}\right\}={\frac {1}{k!}}\sum _{i=0}^{k}(-1)^{i}{\binom {k}{i}}(k-i)^{n}$

1. Ronald L. Graham, Donald E. Knuth, Oren Patashnik (1988) Concrete Mathematics, Addison–Wesley, Reading MA. ISBN 0-201-14236-8, p. 244.

URL copied to clipboard
PLANETCALC, Stirling numbers of the second kind