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



G. Silldorf 78 considered the number f(s, k) of decompositions of s into 

 k integral summands ^ 0, and the number f r (s, k) in which r is the least 

 summand. In the former, occurs in the first place f(s, k 1) times, 1 

 occurs fi(s 1, k 1) times, etc. But/(s, k) = f r (s + rk y k). Hence 



f(s, k} = /(s, k - 1) +/( - k, k - 1) + - - . +/( - rk, k - 1) + 

 Thus/(s, 2) = (s + 2) or %(s + 1) according as s is even or odd, 

 /(s, 3) = (s 2 + 6s + 12)/12, s = (mod 6), 



with similar results for s = 1, , 5 (mod 6). Let F(s, k} be the number of 

 combinations without repetitions of k elements with the sum s. Then 



F(s, k) = F(s - k, k - 1) + + F(s - rk, k - 1) + -, 



[Euler, 9 315]. There are as many partitions in parts ^ m as into m 

 or fewer parts. The number of ways s can be expressed as a sum of numbers 



. 

 ^ m, with repetitions allowed, is 



o _ 



2-i - ( s - m - l) (s - m + 1)2--"^ +(s-m-l) 1 - - 8 ~- 



ft 



, . N s 3w + 3 . 



- (s - m 1) 2 - o - 2 s - 3m ~ 4 + . 



o 



F. Gambardella 79 noted that ax + by + cz = m has 

 %q(2m + a + b + c abcq) + s + k 



sets of integral solutions if a, b, c are positive and relatively prime in pairs, 

 and m>0, m = C7 + X, 7 + != qab + r. Here s is the sum of the 

 quotients and pi, , p r the remainders upon dividing X, X + c, , 

 X + (r l)c by ab; while & is the number of solvable equations ax -f by = p a . 



T. P. Kirkman, 790 counting 5-1 = 1-5 = 1-3 + 1-2 = 1-3 + 2-1= 

 as partitions of 5, evaluated the sum of the reciprocals of (2e 1 ) mi (2e 2 )'" 11 

 wii! m 2 ! , for all such partitions miei + m^ + of R. 



J. J. Sylvester 80 noted that a list of all partitions of n may be checked by 



2(1 - x + xy - xyz + ) = 0, 



summed for all the partitions, where in any partition, x is the number of 

 1's, y the number of 2's, etc. 



Von Wasserschleben 81 expressed 60& as a sum of four numbers each a 

 prime or product of two equal or distinct primes, for k = 1, , 16. 



* L. Jelinek 82 treated a kind of partitions. 



78 Ueber die Zerlegung ganzer Zahlen in Summanden, Progr. Salzwedel, 1870, 17 pp. 

 " Giornale di Mat., 9, 1871, 262-5. Extensions by C. Sardi, 11, 1873, 123. 

 79 Math. Quest. Educ. Times, 15, 1871, 60-3; 16, 1872, 74-5. 



80 Report British Assoc., 41, 1871 (1872), 23-5; Coll. Math. Papers, II, 701-3. 



81 Archiv Math. Phys., 54, 1872, 411-8. 



82 Die Wiirfelzahlen u. die Zerlegung einer Zahl in ganzen Z., deren Summe gegeben ist, 



Progr. Wiener Neustadt, 1874. 



