106 HISTORY OF THE THEORY OF NUMBERS. [CHAP, ill 



Again, by comparing (5) with the corresponding relation with n replaced 

 by n + 1, it is found that 



(N + 1, n + 1, m) = (N, n + 1, m) + (N, n, m) (N m, n, iri). 

 Finally, by expanding (1 x m ) n and (1 x)~ n by the binomial theorem, 



(n + \-l\ (n\(n + \-m-l\ 

 (n + l,n, m ) = ( x )-(,)( x _ w J 



(n\(n + \ -2m-l\ (n\(n + X - 3m - 1\ 

 \2/\ X - 2m / \3/\ X - 3m / 



Euler's proofs were made for m = 6 and, except for the third formula, 

 involve incomplete inductions. By evaluating the coefficient of X N in the 

 expansion of 



(++ x*)(x + + x*)(x + + x 12 ) 



= (x* - x 9 - - - - rc 29 )/(l - z) 3 , 



Euler found the number of partitions of N ^ 26 into three parts, the 

 first part ^ 6, the second ^ 8, the third =5 12. 



As to the problem known as the rule of the Virgins [cf. Sylvester, 54 and 

 note 188 of Ch. II], the number of sets of integral solutions p, q, , 

 each is 0, of the pair of equations 



ap + bq + = n, ap]+ fiq + = v, 

 is the coefficient [not determined] of x n y v in the expansion of 



(1 -x a y a )~ l (l -aV)- 1 --.. 



K. F. Hindenburg 15 gave a method, different from Boscovich's, for 

 listing all partitions of n. For n = 5, the method lists them in the order 



5, 14, 23, 113, 122, 1112, 11111. 



Hindenburg 16 gave a method of listing all partitions of n into m parts. 

 The initial partition contains m 1 units and the element n m + 1. 

 To obtain a new partition from a given one, pass over the elements of the 

 latter from right to left, stopping at the first element / which is less, by 

 at least two units, than the final element [/ = 2 in 1234]. Without 

 altering any element at the left of /, write / + 1 in place of / and every 

 element to the right of / with the exception of the final element, in whose 

 place is written the number which when added to all the other new elements 

 gives the sum n. The process to obtain partitions stops when we reach 

 one in which no part is less than the final part by at least two units. 



Case n = 10, m = 4: 



16 Methodus nova et facilis serierum infinitarum exhibendi dignitates, Leipsae, 1778. Infini- 

 tinomii dignitatum historia, leges, ac formulae, Gottingae, 1779, 73-91 (166, tables of 

 partitions). A less interesting method is given in a Progr., 1795. 



"Exposition by C. Kramp, Elcmens d'Arith. Universelle, 1808, 339. Quoted by Trudi. 98 



