10176

# Lah Numbers

The Lah numbers count the number of ways to partition a list of elements into disjoint, nonempty subsets, where the order of elements in a subset is significant. This is in contrast to the Stirling numbers of the second kind, which count such partitions where the order of elements in a subset does not matter, and Stirling numbers of the first kind, which consider the cycles of a permutation. This Demonstration illustrates the different partitions that Lah numbers count.

### DETAILS

Snapshot 1: counts the number of ways to partition elements into only one list. Because the order of elements in the list is significant, and there are ways to order elements, .
Snapshot 2: Similarly, because there is only one way to partition elements into nonempty subsets (the order of the subsets themselves does not matter).
Snapshot 3: The Lah numbers can be computed recursively; by comparing Snapshot 1 and Snapshot 3, it is apparent that and are related.
Reference
[1] J. Riordan, Introduction to Combinatorial Analysis, New York: John Wiley, 1958 pp. 43–44.

### PERMANENT CITATION

 Share: Embed Interactive Demonstration New! Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details » Download Demonstration as CDF » Download Author Code »(preview ») Files require Wolfram CDF Player or Mathematica.

#### Related Topics

 RELATED RESOURCES
 The #1 tool for creating Demonstrations and anything technical. Explore anything with the first computational knowledge engine. The web's most extensive mathematics resource. An app for every course—right in the palm of your hand. Read our views on math,science, and technology. The format that makes Demonstrations (and any information) easy to share and interact with. Programs & resources for educators, schools & students. Join the initiative for modernizing math education. Walk through homework problems one step at a time, with hints to help along the way. Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet. Knowledge-based programming for everyone.