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.

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.