680 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1951 



must be greater than the number of ways N different objects can be placed 

 in A different boxes (labelled 1, • • • ^ A); i.e. 



A"" <Q{N,A)A\<<i>{N)A\. 



To obtain the best lower bound from (12) one may maximize on A. The 

 best value of A to use is one which comes close to satisfying 



N-l 



(' - r - ^ 



For large N the solution is approximately 



A= ^ 



logeN 



To perform the minimization in theorem IV more carefully one would 

 solve 



Aq loge ^0 = N 



for Aq . This is the minimizing equation given in theorem IV with 

 To = loge ^c . It is also very nearly the minimizing equation given in 

 theorem V. 

 Then our proofs of thegrems IV and V show that 



iN Aq—I 



Aol - -^ ^ - -(loge^o)^* 

 For large N these bounds differ by a factor of about 



gTT N 



e Vloge TV 



More accurate information about the behavior of 0(iV) for large N is 

 provided by an asymptotic series found by L. F. Epstein.^ The first term in 

 his series is 



Figure 6 is a graph of <^(A^) vs. N using a log log scale for <i>{N). The 

 points are exact values and the curves show the upper and lower bounds. 



» L. F. Epstein, J.M.P., 18, 3, pp. 153-173 (1939). 



