Stirling number of second kind
Clinical significance : the \ (n-\) th differentiated into pellets \ (m \) th box without distinction, and the number of empty cassette without a second embodiment of the Stirling number of classes, referred to as \ (S (n-, m) \) .
Recursive formula : \ (S (n-, m) = m * S (n--. 1, m) + S (n--. 1,. 1-m) \)
Clinical significance demonstrated: the pellets into a new box , can be selected into the \ (m \) one by one in a box, it can also be put into a new box.
计算公式: \(S(n,m) = \frac{1}{m!} \sum_{i=0}^{m} (-1)^i \binom{m}{i} (m-i)^n\)
Corollary 1 : when (n <m \) \ time, \ (\ sum_ I = {0} ^ {m} (-1) ^ I \ Binom {m}} I {(mi The) n-^ = 0 \) .
Clinical significance proof: when (n <m \) \ time, no matter how empty boxes are placed, so \ (S (n-, m) = 0 \) , since \ (\ frac {1} \ ge {m!} 0 \) , so \ (\ sum_ I = {0} ^ {m} (-1) ^ I \ Binom {m}} I {(mi The) n-^ = 0 \) .
Corollary 2 : \ (! \ Sum_ I = {0} ^ {m} (-1) ^ I \ Binom {m}} I {(mi The) m ^ m = \)
Clinical significance demonstrated: \ (S (m, m) =. 1 \) , so \ (\ frac {1} { m!} \ sum_ {i = 0} ^ {m} (-1) ^ i \ binom {m} {i} (mi) ^ n = 1 \) , on both sides of the equation by a same \ (m! \) to