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



linear inequalities connected with a number of linear relations. Such 

 theories are grouped under the title " partition analysis." As regards 

 the simple partition of numbers the idea results in laying foundations deeper 

 than the intuitive generating functions which served Euler and his successors 

 as points of departure. . There is an extension in the direction of two dimen- 

 sions in such wise that the parts are laid out in the compartments of a chess 

 board of any dimensions, a partition being defined as a distribution of 

 numbers such that in every row and in every column of the board a de- 

 scending order of part magnitude is in evidence. The complete enumerative 

 solution of this question for a complete or incomplete lattice or chess board 

 is reached. The solution depends upon the idea of a lattice permutation 

 and of an associated lattice function. An assemblage of letters a^af "" 

 is said to be a lattice assemblage when the repetitional exponents satisfy 

 the condition a\ ^ a z = = a s ; and of this assemblage a permutation 

 is a lattice permutation if the first k letters (k being any number < s) 

 of the permutation constitute a permutation of a lattice assemblage 

 af'aa] 2 af 8 - These permutations have been enumerated, but the theory 

 of the derived lattice functions is not yet complete. The theory of parti- 

 tions in three dimensions is completed in this book only as far as the simplest 

 case when the parts are placed at the angular points of a single cube. The 

 enumeration of the partitions of multipartite numbers is investigated 

 principally by means of J. Hammond's 217 " differential operators [Mac- 

 Mahon 176 ]. The problem of enumerating partitions which do not involve 

 sequences of parts is considered in Vol. I. 



* L. von Schrutka 218 gave an extended account of methods employed 

 to further develop Vahlen's 150 results. 



R. Goormaghtigh 219 noted that, if N is the sum of the consecutive integers 

 comprised between v + 1 and n, then 2N = (n v}(n + v + 1) and the 

 number of couples n, v is the number of odd divisors > 1 of N. 



G. H. Hardy and S. Ramanujan 220 proved that the logarithm of the 

 number p(ri) of partitions of n is asymptotic to 7rV2n/3, and the logarithm 

 of the number of partitions of n into distinct positive integers is asymptotic 

 to TT Vn/3. They 221 developed a general method for the discussion of these, 

 and analogous problems of combinatory analysis, by means of the methods 

 of the theory of functions of a complex variable. This method is, within 

 limits, applicable to the study of all numerical functions which occur as 

 coefficients in power series possessing the unit circle as a natural boundary. 

 In this particular problem it leads to the result that 



. , Id 

 P< = : - T- 



- 1/24 



2170 Proc. London Math. Soc., 13, 1882, 79; 14, 1883, 119. 



218 Jour, fur Math., 146, 1915-6, 245-254. Sitzungsber. Akad. Wiss. Wien (Math.), 126, Ila, 



1917, 1081-1163. 



219 L'interm6diaire des math., 24, 1917, 95. 



220 Proc. London Math. Soc., (2), 16, 1917, 131. 



221 Comptes Rendus Paris, 164, 1917, 35-38. Proc. London Math. Soc., (2), 17, 1918, 75-115. 



