First class Stirling number and second class Stirling number __ theory

First class Stirling number S (p,k)

A combinatorial interpretation of S (p,k) is the number of methods that arrange P objects into K-non-empty loops.

Recursive formula for S (p,k): s (p,k) = (p-1) *s (p-1,k) +s (p-1,k-1), 1<=k<=p-1

Boundary conditions:s (p,0) =0, p>=1 S (p,p) =1, p>=0

Description of recurrence Relationship:

Considering the P item, p can constitute a non-empty loop arrangement, so that the former p-1 items constitute k-1 loops, and the method number is S (p-1,k-1);

It is also possible for the former p-1 to form a K-Non-empty loop, while the P-item is inserted to the left of the first item, which has the (p-1) *s (p-1,k) method. Second Class Stirling number S (p,k)

A combinatorial interpretation of S (P,K) is the number of methods that divide p objects into K-Non-empty, indistinguishable (which can be understood to be no number of boxes).

S (p,k) is the number of methods that separate p individuals into k-rooms (e.g., marked with room numbers) (no vacancies).

The recursive formula for S (p,k) is:s (p,k) =k*s (p-1,k) +s (p-1,k-1), 1<= k<=p-1

Boundary conditions:s (p,p) =1, p>=0 S (p,0) =0, P>=1

Description of recurrence Relationship:

Considering the P item, p can constitute a non-empty set, at which point the former P-1 object constitutes a k-1, Non-empty set, with the method number S (p-1,k-1);

It is also possible to form a K-non-empty set of p-1 items, and the P items are placed in any one, so that there is a k*s (p-1,k) method.

The first kind of Stirling number has the same initial condition as the second kind of Stirling number, but the recurrence relation is different.