PREFACE. vii 



of ITj=r(l x y ) is the generating function for the number of unrestricted 

 partitions of n, where now also 5 and 4+1 are counted. Again, the number 

 of partitions of n into m or fewer parts ^=t is the coefficient of x n in the 

 expansion of 



where D is the above product. Euler stated empirically the important 

 fact that 



A-=l 



which has since been proved by many writers, in particular by Jacobi in 

 his Fundamenta Nova of 1829, where he made important applications of 

 elliptic functions to the theory of partitions. As noted by Legendre in 1830, 

 the last formula implies that every number, not a pentagonal number 

 (3n 2 n)/2, can be partitioned into an even number of distinct integers as 

 often as into an odd number, while (3n 2 n)/2 can be partitioned into an 

 even number of parts once oftener or once fewer times than into an odd 

 number of parts, according as n is even or odd. Jacobi in 1846 extended 

 this result to partitions into any given distinct elements. 



In 1853 Ferrers gave a diagram which establishes a reciprocity between 

 the partitions of the same number. The partition 3+3+2+1 is repre- 

 sented by four rows of dots containing 3, 3, 2, 1 dots, respectively, such that 

 the left-hand dots are in the same vertical column. Reading the diagram 

 by columns, we get the partition 4+3+2. 



Sylvester stated in 1857 that the number of partitions of n into given 

 positive integral elements a\, . . ., a r with repetitions allowed is 2TF g , where 

 the "wave" W q is the coefficient of l/t in the development in ascending 

 powers of t of 



the summation extending over the various primitive qih roots p of unity. 

 Proofs were soon given by Battaglini, Brioschi, Roberts, and Trudi; 

 Sylvester published his own method in 1882. Cayley wrote several papers 

 on the theory and its applications. 



During the years 1882-84, Sylvester and his pupils at Johns Hopkins 

 University published many papers on partitions, in particular on their 

 graphical representation, with the aim to derive the chief theorems con- 

 structively without the aid of analysis. 



Beginning with his paper of 1886 on perfect partitions, Major MacMahon 

 has made numerous contributions to the thoery of partitions and the more 

 general subject of combinatory analysis, culminating in his treatise in two 

 volumes published in 1915-16 (see the report, pp. 161-2). 



Vahlen proved in 1893 that, among the partitions of s into distinct parts 

 the sum of whose absolutely least residues modulo 3 equals a given integer h, 

 there occur as many partitions into an even number of parts as into an odd 

 number of parts, except only when s is the pentagonal number (3h 2 h)/2, 



