CHAP, ill] PARTITIONS. 103 



Hence a = 51, = n$8, y = n 3 (5, 5 = n 6 ), , where 1, 3, 6, are the 

 successive triangular numbers. From the above series for a, (3, , we 

 see that 



m w = 



Hence m w is also the number of ways m n can be obtained by addition 

 from 1, -,/* The former recursion formula for Wi (M) gives 



(2) m (M) = (m - M) (M) + (m - l)^". 

 He stated, as a fact he could not prove, 



(3) p(x) = 



and that the reciprocal of the product is 1 + x + 2x 2 + 3z 3 + 5x 4 + , 

 the coefficient of x s being the number of ways s can be partitioned into 

 equal or distinct parts. As to (3), see Euler 1 " 6 , Ch. X, Vol. 1. 



Euler, 4 in a letter to N. Bernoulli, Nov. 10, 1742, stated the preceding 

 facts on partitions. The answer to the second problem he stated in the 

 following equivalent form: m w is the coefficient of n m in the expansion of 

 n"/{(l -ri)(l -n 2 )---(l - Of. 



Euler 5 gave (3) and p(x) = 1 - PI + P 2 - P 3 + [see Euler 9 ]. 



P. R. Boscovich 6 gave a method of finding all the partitions of a given 

 number n into integral parts > 0. Write down n units in a line. Replace 

 the last two units by 2, then replace two units by 2, etc. Next, write 

 n 3 units and 3; replace two units by 2, etc. Then write n 6 units 

 and two 3's; replace two units by 2, etc. Thus the partitions of 5 are 



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



He applied partitions to find any power of a series in x, also in a paper, 

 ibid., 1748. In his third paper, ibid., 1748, he showed how to list the parti- 

 tions of n into parts ^= m, by stopping his above process just before a part 

 m + 1 would be introduced. He applied the rule also to the case when the 

 parts are any assigned numbers. He treated the problem to find all the 

 ways in which a given integer n can be decomposed in an assigned number 

 m of parts, equal or distinct; but the solution by Hindenburg 16 is much 

 more simple and direct. Boscovich attempted in vain to find a formula 

 for the number of partitions. He gave elsewhere 7 his rule. 



K. F. Hindenburg 8 would obtain the partitions of 8 by annexing unity 

 to those of 7, and supplement them with 



2222, 224, 233, 26, 35, 44, 8. 



4 Opera postuma, 1, 1862; Corresp. Math. Phys. (ed., Fuss), 2, 1843, 691-700. 

 6 Letter to d'Alembert, Dec. 30, 1747; Bull. Bibl. Storia Sc. Mat., 19, 1886, 143. 



6 Giornale de' Letterati, Rome, 1747. Extract by Trudi, 98 pp. 8-10. 



7 Archiv der Math, (ed., Hindenburg), 4, 1747, 402. 



8 Ibid., 392, and Erste Samml. Coinbinatorisch-Analyt. Abhand., 1796, 183. Quoted from 



G. S. Klugel's Math. Worterbuch, 1, 1803, 456-60 (508-11, for references). 



