84 BELL SYSTEM TECHNICAL JOURNAL 



and 



ci , C2 , • • • , c„ = (1, 2, 4, ■ • • , 2''-i) 



We may assume the a, arranged in a non-decreasing sequence, Oi < C2 < 

 di < ■ ■ ■ < Cn- In case ai = 2 the proof is easy. We have 



1, 2, 4, • • • , 2-1 B 



1, 2, 4, • • • , 2-1 C 



2,4, 8, •••,2" A 



and a flow has occurred in the set 



4, 8, 16, • ■ • , 2" 



to give a2 , az , ■ ■ ■ , an ■ Now any permissible flow in C corresponds to a 

 permissible flow in either A or B since if 



Cj = aj + bj > Ci — ai -\- bi 



then either a, > ai or bj > bi 



Thus at each flow in the sum we can make a corresponding flow in one or the 

 other of the summands to keep the addition true. 

 Now suppose Qi > 2. Since the a, are non-decreasing 



(« - 1) 02 < (2"+i - 2) - ai < 2«+i - 2 - 3 

 Hence 



a,-\< i -^ - 1 < 2"-^ 



n — 1 



the last inequality being obvious for n > 5 and readily verified for n < 5. 

 This shows that (oi — 1) and (c2 — 1) lie between some powers of two in the 

 set 



1,2,4, ••• ,2"-i 

 Suppose 



2-^-1 < (ci - 1) < 2« 



2P-1 < (02 - 1) < G" q< P<{n- !)• 



Allow a flow between 2' and 2«-i until one of them reaches (ai — 1), the 

 other (say) R\ similarly for (02 — 1) the other reaching S. As the start 

 toward our decomposition, then, we have the sets (after interchanges) 



L 



{a, - 1) 1 



1 02-1 



2, 4 • • • 2""^ R 2'^^ • • • 2""^ 2" 2"^^ • ■ • 2"~^ 

 2, 4 • • • 2'"" 2""^ 2' ■■■ 2''~^^S 2"^^ ■ ■ ■ 2""^ 



fli 



4, 8 ■ • • 2'"' • • • 2"^^ • • • 2" 



L 



