148 HISTORY OF THE THEORY OF NUMBERS. [CHAP, in 



KB direction to the AK direction. The number of different lines of route 

 with exactly s essential nodes is (*)(*). Each of these lines of route 

 represents 2 p+q ~ s ~ l compositions. For tripartite numbers, we need three 

 dimensions. Generating functions were found for the number of all 

 compositions of multipartite numbers; he 155 - 194 treated this topic also 

 later. 



K. Zsigmondy 156 partitioned m into distinct parts each unity or a product 

 of distinct ones of the first s primes; for example, the parts may be 1, 2, 3, 

 5, 2-3, 7, 2-5, 11. If the partition has an even number of parts, consider 

 the excess of E of the number of parts with an odd number of prime factors 

 over the number of terms with an even number of prime factors, unity 

 being a possible term. Thus for 11 = 2-3 + 5, E = 0; for 2-5 + 1, 

 E = - 2; for 5 + 3 + 2 + 1, E = 2. But if the partition has an odd 

 number of parts, let E be the excess of the number of parts with even 

 over that with odd number of prime factors. Thus for 2-3 + 3 + 2, 

 or 7 + 3 + 1 or 1 1, E = 1. The sum 2 n of the E's for these 6 partitions 

 of 11 is 0-2 + 2-1-1-1 = - 3. Next, <r u = 3 - 3 = is the 

 excess of the number of the partitions of 11 into an odd number of parts 

 over those into an even number of parts. He proved that, if m > 1, 

 2 m + a m -\ = 1 or 0, according as m is the (s + l)th prime p or is < p. 

 For example, if p = 13, m = 11, we had S OT = 3, while <r m _i = 3 since 

 the partitions of 10 into an odd number of parts are 2-5, 7 + 2 + 1, 

 2-3 + 3 + 1 and 5 + 3 + 2, while 7 + 3 is the only partition into an 

 even number of parts. 



W. J. C. Sharp stated and H. J. Woodall 156 " proved that, if P n is the 

 number of partitions of n without repetitions and Q n is the number of par- 

 titions into odd parts, then P n = Qn + Qn-zPi + Qn-tP* + > and that 

 the same formula holds when P n and Q n denote the number of such parti- 

 tions with repetitions. 



L. Eamonson 1566 expressed the number of partitions of 2n into two 

 primes in terms of the number of odd primes =i k for various values 

 of k. 



L. J. Rogers 1560 established the important identities 



, 



n=l 



165 Phil. Trans. Roy. Soc. London for 1894, 185, A, 111-160. 



168 Monatshefte Math. Phys., 5, 1894, 123-8. 



1660 Math, quest. Educ. Times, 60, 1894, 41. 



1666 Ibid., 63, 1895, 116-7. 



J 66c Proc. London Math. Soc., (1), 25, 1894, 328-9, formulas (1), (2). Cf. papers 226-8. 



