TV^-TERMINAL SWITCHING CIRCUITS 679 



A switching function represents one of these 0(iV) different states for 

 each of the l" different switch settings. Hence the number of switching 

 functions is 



Although there is no simple formula for 0(iV), a generating function for 

 0(A^) is well known :^ 



(8) ^..-1^ |^0W^n_- 



A recursion formula which can be used to calculate 0(7V) is 



(9) 0(iv + i) = i:c^..<^w. 



A;=0 



When N is large <^(iV") can be estimated with the help of the upper and 

 lower bounds to be derived. These bounds will be of use to us later mainly 

 because they show that, for large iV, log <^(iV") is approximately iV" log TV. 



Theorem IV . 



(10) *(^) ^ -e 



^\ gAT/logeJV 



L'"^' i^l 



Proof. The maximum value of | e^* ^ | on the circle | z | = r is e***^ ^ . Using 

 (8) and Cauchy's inequaUty for the iV*^ coefl&cient in a power series, 



(11) <t>{N) < 



N\e''-' 



for all r > 0. The best estimate of 4>{N) will be obtained by minimizing (11) 

 on r. To do this one sets r = ro where 



roe' = N. 



The simpler result (10) is obtained from (11) by setting 



Theorem V. 



(12) TT — *^^^^ f^^ ^^^ integers A. 



Proof. Let Q{Nj A) he the number of ways that N different objects can 

 be distributed into 1,2, • • • , or ^ indistinguishable parcels. Then Q{N, A) A ! 



4 W. A. Whitworth, Choice and Chance, p. 88, Cambridge, Bell, 1901. 



