CHAPTER III. 



PARTITIONS. 



G. W. Leibniz 1 asked Bernoulli if he had investigated the number of 

 ways a given number can be separated into two, three or many parts, 

 and remarked that the problem seemed difficult but important. Leibniz 2 

 used the term number of divulsions for the number of ways a given integer 

 can be expressed as a sum of smaller integers, as 3, 2 + 1, 1 + 1 + 1, 

 and noted the connection with the number of symmetric functions of a 

 given degree, as Za 3 , Za 2 6, Za&c. 



L. Euler 3 found relations between A = Za, B = Za 2 , C = Za 3 , -, and 



a Za, /3 = Za&, 7 = Za&c, , 



SI = Za, 



93 = a 2 + ab + 6 2 + ac + , 



< = a 3 + a 2 6 + a& 2 + 6 3 + a 2 c + a&c + , 



) = a 4 + a?b + + a&cd + , 



We have 



P = 2^- = Az + z 2 + Cz 3 + -, 

 1 az 



+ Cz* ---- 



Rdz 1 + az 

 R = H(l + az} = 1 + az + /3z 2 + 



= RQ. 



az 



Hence 



A = a, aA - B = 2$, PA - aB + C = 87, 



Next, expanding (1 + az)~ l , we get 



/t 



Now take a = n, b = ri*, c = n 3 , . Then 



A = n/(l - n), 5 = n 2 /(l - w 2 ), 

 Hence 



nz ri*z nz tfz 2 



P = 3 -- r- ^ - T -r " T~ ' +1 -- 2+ '"> 

 I nz I ri*z I n 1 n z 



R= (1+ nz)(l + tfz) = 1 + az + /5z 2 + -, 

 a = n + n 2 + n 3 + , 



1 Math. Schriften (ed., Gerhardt), 3, II, 1856, 601; letter to Job. Bernoulli, 1669. 



2 MS. dated Sept. 2, 1674. Cf. D. Mahnke, Bibliotheca Math., (3), 13, 1912-3, 37. 



1 "Observ. anal, de combinationibus," Comm. Acad. Petrop., 13, ad annum 1741-3, 1751, 

 64-93. 



101 



