The classical Stirling Numbers of the Second Kind, denoted *S(n, k)*, are the number of ways to partition *n* distinct objects into exactly *k* nonempty subsets. The numbers presented here were calculated using the well-known recurrence

*S(n, k) = S(n - 1, k - 1) + k * S(n - 1, k)*.

If we let the *n* objects be the integers 1, 2, ..., *n*, we can define a generalization of the Stirling Numbers of the Second Kind. Define the Reduced Stirling Numbers of the Second Kind, denoted *S ^{d}(n, k)*, to be the number of ways to partition the integers 1, 2, ...,

*n*into exactly

*k*nonempty subsets such that all elements in each subset have pairwise distance at least

*d*. That is, for any integers

*i*and

*j*in a given subset, we require |

*i*-

*j*| ≥

*d*. It has been shown that these numbers satisfy

*S ^{d}(n, k) = S(n - d* + 1,

*k - d*+ 1), where

*n ≥ k ≥ d*

(hence the name "reduced"). Observe (both by definition and by the reduction formula) that *S ^{1}(n, k) = S(n, k)*, the classical Stirling Numbers of the Second Kind.

Available for download are the output file (.txt 43mb) containing the values of *S(n, k)* for 1 ≤ *k* ≤ *n* ≤ 500 and the Java source code that generated the output file.