CHAP. Ill] 



PARTITIONS. 



151 



is a set of solutions of 



(Xa 



(\d 



= Xra 



Conversely, if X, ju are suitably chosen positive integers, every set of solu- 

 tions ^ of the latter is a set of solutions of the pair of equations. 



K. Glosel 166 considered the number C r (<r) of ways of expressing a as a 

 sum of r distinct positive integers, gave a new proof of De Morgan's 28 

 formulas for r = 2, 3, and, for r = 4, simpler expressions than Zuchris- 

 tian's. 149 If {a} is the integer nearest to a, 



Ct(2k + 1} = 



2k(k - 3) 2 

 36 



(2k - 3) (k - 3) 2 

 36 



which may be combined into 



The complicated expression for C 6 (o-) was simplified on page 290. 



P. A. MacMahon 167 gave a report on combinatory analysis and parti- 

 tions. He suggested (pp. 30-1) a method of enumerating multipartite 

 partitions. 



MacMahon 168 noted that a partition (pi- - -p 5 ) has the " separations ' 

 (l>iP2)(paP*)(P&)> (Pipzp^faps), etc., the numbers in any parenthesis being 

 considered as a partition with those parts. It is easily proved that the 

 number of separations of the partition (p?pZ*- )> where in indicates the 

 number of repetitions of the part p i} is identical with the number of parti- 

 tions of the multipartite number TTITTV . Sylvester's method of graphical 



representation of partitions can not be simply extended to multipartite 



J>- 



CL 



D 



C 



partitions. But there is a correspondence between m-partite partitions 

 and (m + 1) -partite compositions. For example, let m = 1 and consider 

 the graph of the bipartite number 76. Each composition has a line of 

 route through the lattice [as MacMahon 154 ], a, b, c being the essential 

 nodes of the line of route shown in the figure. The principal composition 



166 Monatshefte Math. Phys., 7, 1896, 133-141. 



167 Proc. London Math. Soc., 28, 1896-7, 5-32. 



168 Phil. Trans. Roy. Soc. London, for 1896, 187, A, 1897, 619-673. 



Memoir I on Partitions. 



