CHAP, ill] PARTITIONS. 153 



P. A. MacMahon 172 found the number of ways n is a sum of 8 numbers 



rtiz m 4 , 



two solutions being identified if one can be derived from the other by a 

 permutation of the two rows or of the four columns. This question of bi- 

 partition is solved also when the number of columns is arbitrary. 



H. Wolff 173 evaluated the number F^ri) of partitions of n into ^ positive 

 integers #,- arranged in order of magnitude, X Q ^ #1 ^ x 2 = = # M _i, 

 and proved that 



where the summation extends over all decompositions /* = / + gv\ + , 

 while, for each, <j>(n) is the number of partitions of n into/ sets of successive 

 equal parts, followed by g sets of 77 successive equal parts, etc., the various 

 groups not being arranged according to the magnitude of the parts. Thus, 

 for example, n=4=0+0+2+2 and 2 + 2 + + are counted as 

 distinct in computing $(n). 



The number of decompositions of n into X equal parts is evidently 1 

 or according as n is or is not divisible by X, and hence is 



if jR(w/X) denotes the least positive remainder on the division of n by X. 

 If X is the g.c.d. of , 77, , the above 0(w) equals the product of p(n, X) 

 by the number 0(w/X) of partitions n = //X + gr]/\ + . Again, the 

 number of decompositions n = / + gtj is p(n, 77) + C n '/] ~~ L n 'n'l r l']> ^ 

 , 77 are relatively prime and 77' 77^' = =F 1. Recursion formulas for the 

 0's are found and the F M (n) evaluated for /* ^ 6 as explicit functions of n. 

 By means of Bernoullian functions, F M (n) is expressed as a polynomial in 

 n whose coefficients are linear functions of the coefficients of F M _i(n). 



* G. Csorba 173a made an addition to the theory of partitions. 



P. A. MacMahon 174 generalized the concept of a partition into parts 

 ori, 2 , , <x s by replacing the conditions ai ^ 2 = by the conditions 



A? ai + ASPaz + + A?a, ^0 (i = 1, -, r), 



where at least one of the integers A is positive. There is a finite number of 

 fundamental solutions (ctf\ , a c /0 for j = 1, , m of these conditions, 

 such that every solution is of the form = Xio^ 11 + + Xmc4 m) for i = 1, 

 , s, where the X's are positive integers. 



MacMahon 175 treated the generating functions for the enumeration of 

 three-dimensional graphs possessing either ^-symmetry (when each layer 



172 Bull. Soc. Math. France, 26, 1898, 57-64; M. d'Ocagne, p. 16, for n = 3, 4. 



173 tiber die Anzahl der Zerlegungen einer ganzen Zahl in Summanden, Diss., Halle, 1899. 

 1730 Math, es termed ertesito (Hungarian Acad. Sc.), 17, 1899, 189. 



174 Phil. Trans. Roy. Soc. London, 192, A, 1899, 351-401. Memoir II on Partitions. 

 176 Trans. Cambridge Phil. Soc., 17, 1899, 149-170. 



