(Mn) represents the number of solutions that put n different earths into m identical boxes (there is no empty box), and (mn) is the second type of Stirling number (_m^n) represents the number of n different earths Enter the number of schemes in m identical boxes (without empty boxes), and call (_m^n) the Stirling number of the second type （mn) Table shows the n th not different manner ball put into the m th phase with the cartridge in the side case a number of ( not have empty box ) , known as (mn) Is first dicarboxylic Class S T I R & lt L I n- G Number
Properties of Stirling Number
（0 n） = 0 ， （1 n） = 1 ， （2 n） = 2 n - 1 - 1,……, （n - 1 n） = cn 2, （nn） = 1 （_0 ^ n） = 0 ， （_ 1 ^ n） = 1 ， （_ 2 ^ n） = 2 ^ {n-1} -1, ……, （_ {n-1} ^ n） = c_n ^ 2, （_ n ^ n） = first （0n）=0，（1n）=1，（2n）=2n−1−1,……,（n−1n）=cn2,（nn）=1
Recurrence formula: (mn) = m (mn − 1) + (m − 1 n − 1) (Interpretation from the principle of multiplication and addition: Take out a ball and put it in a separate designated box (put the others in m − 1 box) or other boxes with balls) Recursion formula: (_m^n)=m(_m^{n-1})+(_{m-1}^{n-1} ) (Interpretation from the principle of multiplication and addition: \\Take out a ball and put it in a separately designated box (others in m-1 boxes) or put them in other boxes with balls)Handed push public ceremony:（mn）=m（mn−1）+（m−1n−1) ( From the multiplication method original rationale and plus method original management solution read :Take out a th ball is put into a th single separate finger specified in the cartridge sub ( which he is put to m−1 Ge box child inside ) or put it he had the ball in the box Sub years )
Example: How many different equivalence relations are there on the set A = {A, B, C, D }? Example: How many different equivalence relations are there on the set A=\{A,B,C,D\}?Example : Set engagement A={
A,B,C,D } on with a plurality less one is not different in other monovalent off line ?
(1 4) + (2 4) + (3 4) + (4 4) = 15 (_1^4)+(_2^4)+(_3^4)+(_4^4)=15（14）+（24）+（34）+（44）=15